A Phase I/II Study to Evaluate the Safety of Cellular Immunotherapy Using Autologous T Cells Engineered to Express a CD20-Specific Chimeric Antigen Receptor for Patients With Relapsed or Refractory B Cell Non-Hodgkin Lymphomas

Inclusion Criteria:

* Patients must have B-cell non-Hodgkin lymphoma or chronic lymphocytic leukemia/small lymphocytic lymphoma. Eligible lymphoma subtypes include (but not limited to): mantle cell, follicular, lymphoplasmacytic, marginal zone, transformed indolent B cell lymphoma (including transformed chronic lymphoid leukemia \[CLL\]), or diffuse large B cell lymphoma that has relapsed after a response to at least one prior therapy regimen or is refractory to prior therapy; patients with mantle cell lymphoma must have previously been treated with a Bruton tyrosine kinase (BTK) inhibitor and have either had disease progression, intolerance, or exposure to the drug for at least 3 months; patients with CLL/SLL are eligible if they had disease progression or intolerance to BTKis and/or a BCL-2 inhibitors; they are also required to have been treated with the other agent for at least 3 months (i.e. patients with progression/intolerance to BTKi need to be treated with a BCL-2 inhibitor for at least 3 months, and patients with progression/intolerance to BCL-2 inhibitor need at least 3 months of exposure to a BTKi); patients with de novo diffuse large B-cell lymphoma (DLBCL) must meet one of the following criteria:

  * Biopsy-proven refractory disease after a frontline regimen containing both an anthracycline and rituximab or other anti-CD20 antibody (i.e. "primary refractory"), where any disease recurring within 6 months of completion of the regimen is considered refractory
  * Relapsed or refractory disease after at least one of the following:

    * At least 2 lines of therapy (including at least one with an anthracycline and anti-CD20 antibody)
    * Autologous stem cell transplant
    * Allogeneic stem cell transplant
* Patients with large cell lymphoma transformed from indolent lymphomas are eligible if previously treated with anthracycline containing regimen for either the indolent or large cell histology
* Patients with central nervous system (CNS) lymphoma need to meet one of the following criteria:

  * Primary CNS lymphoma:

    * Progressive disease after 3 cycles or an inadequate response after at least 4 cycles of a high-dose methotrexate (MTX) containing regimen in the opinion of the treating physician, OR
    * Not eligible for MTX therapy due to commodities or tolerance issues per treating physician OR
    * Recurrent disease after an initial response to MTX-based treatment
  * Secondary CNS lymphoma:

    * An inadequate rtesponse to at least 2 cycles of a MTX containing regimen, OR
    * Recurrent disease after an initial response to MTX-based treatment
* Patients must be 18 years of age or older, of any gender, race or ethnicity
* Patients must be capable of understanding and providing a written informed consent
* Negative serum pregnancy test within 2 weeks before enrollment for women of childbearing potential, defined as those who have not been surgically sterilized or who have not been free of menses for at least 1 year
* Fertile male and female patients must be willing to use an effective contraceptive method before, during, and for at least 4 months after the CAR T cell infusion
* Patients must have a Karnofsky performance status of \>= 60%
* Confirmation of diagnosis by internal pathology review of initial or subsequent biopsy or other pathologic material at Fred Hutchinson Cancer Research Center (FHCRC)/Seattle Cancer Care Alliance (SCCA)/University of Washington (UW)/Harborview Medical Center (HMC)
* Evidence of CD20 expression by immunohistochemistry or flow cytometry on the tumor specimen obtained with the biopsy performed with screening; if the CD20 expression on the screening tumor biopsy is unclear or could not be assessed due to technical reasons, CD20 expression on a concomitant tumor specimen (such as marrow biopsy or circulating tumor cells) may be used to satisfy this requirement. For CLL and lymphoplasmacytic lymphoma/Waldenström macroglobulinemia (WM) patients, the screening tumor biopsy can we waived for clinical reasons after discussion with the study principal investigator (PI). For patients with CNS lymphoma, a screening tumor biopsy is not required and evidence of CD20 expression can be documented from the original or prior biopsies
* Serum creatinine =\< 2.5
* Total bilirubin =\< 3.0 mg/dL
* Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) =\< 5 x the upper limit of normal
* Adequate pulmonary function, defined as =\< grade 1 dyspnea and saturated oxygen (SaO2) \>= 92% on room air; if pulmonary function test (PFT)s are performed based on the clinical judgment of the treating physician, patients with forced expiratory volume in 1 second (FEV1) \>= 50% of predicted and carbon monoxide diffusing capability (DLCO) (corrected) of \>= 40% of predicted will be eligible
* Adequate cardiac function, defined as left ventricular ejection fraction (LVEF) of \>= 50% as assessed by echocardiogram or multigated acquisition (MUGA) scan, or LVEF of 45-49% and clearance by a cardiologist
* Measurable disease that can be accurately measured in at least one dimension as \>= 1.5 cm with computed tomography (CT), ultrasound, or magnetic resonance imaging (MRI) techniques. Extranodal disease that is measurable by fludeoxyglucose F-18 (FDG)-positron emission tomography (PET) imaging only will also be allowed. Note that if an excisional biopsy was performed that removed the sole site of measurable disease, the patient will not be eligible for leukapheresis and generation of CAR T cell product
* For patients with CNS lymphoma, a lesion \>= 1cm on brain or spine MRI is required
* Patients with waldenstrom macroglobulinemia (WM) without radiologic evidence of disease are eligible if they have measurable disease, as defined by serum monoclonal M-spike of \>= 0.5 g/dL
* ELIGIBILITY FOR LYMPHODEPLETION CHEMOTHERAPY: Absence of uncontrolled active infection (bacterial, fungal, viral, mycobacterial) not responding to treatment with antibiotics, antiviral agents, or antifungal agents
* ELIGIBILITY FOR LYMPHODEPLETION CHEMOTHERAPY: Absence of active autoimmune disease requiring ongoing systemic immunosuppressive therapy
* ELIGIBILITY FOR LYMPHODEPLETION CHEMOTHERAPY: Negative serum pregnancy test within 2 weeks before lymphodepletion chemotherapy for women of childbearing potential, defined as those who have not been surgically sterilized or who have not been free of menses for at least 1 year
* ELIGIBILITY FOR LYMPHODEPLETION CHEMOTHERAPY: No treatment with any investigational agent on a different clinical trial between enrollment and lymphodepleting chemotherapy
* ELIGIBILITY FOR LYMPHODEPLETION CHEMOTHERAPY: Serum creatinine =\< 2.5
* ELIGIBILITY FOR LYMPHODEPLETION CHEMOTHERAPY: Total bilirubin =\< 3.0 mg/dL
* ELIGIBILITY FOR LYMPHODEPLETION CHEMOTHERAPY: AST and ALT =\< 5 x the upper limit of normal
* ELIGIBILITY FOR LYMPHODEPLETION CHEMOTHERAPY: Adequate pulmonary function, defined as =\< grade 1 dyspnea and SaO2 \>= 92% on room air; if PFTs are performed based on the clinical judgment of the treating physician, patients with FEV1 \>= 50% of predicted and DLCO (corrected) of \>= 40% of predicted will be eligible
* ELIGIBILITY FOR LYMPHODEPLETION CHEMOTHERAPY: Adequate cardiac function, defined as left ventricular ejection fraction (LVEF) of \>= 50% as assessed by echocardiogram or MUGA scan, or LVEF of 45-49% and clearance by a cardiologist; if subject receives cardiotoxic chemotherapy after enrollment, repeat echocardiogram or MUGA is required to reestablish eligible LVEF
* ELIGIBILITY FOR LYMPHODEPLETION CHEMOTHERAPY: Patients must have a Karnofsky performance status of \>= 60%. Patients with CNS lymphoma with KPS of \>= 50% are eligible if performance status is low because of the active lymphoma
* ELIGIBILITY FOR LYMPHODEPLETION CHEMOTHERAPY: Measurable disease that can be accurately measured in at least one dimension as \>= 1.5 cm with CT, ultrasound, or MRI techniques; extranodal disease that is measurable by FDG-PET imaging only will also be allowed; note that if an excisional biopsy was performed that removed the sole site of measurable disease, the patient is not be eligible for lymphodepletion and CAR T cell infusion; measurable disease can be based on the imaging study done during the screening unless the patient received treatment in the interim, in which case imaging should be repeated. For patients with CNS lymphoma, a lesion \>= 1 cm on the brain or spine MRI is required. Patients with WM without radiologic evidence of disease are eligible if they have measurable disease, as defined by serum IgM level \>= 0.5g/dL
* ELIGIBILITY FOR LYMPHODEPLETION CHEMOTHERAPY: Patients must require no corticosteroid therapy or dose of less than 15 mg per day of prednisone or the equivalent; pulsed corticosteroid dose for disease control is acceptable until the day before the start of lymphodepletion
* ELIGIBILITY FOR LYMPHODEPLETION CHEMOTHERAPY: Patients must have no active acute or chronic GVHD

Exclusion Criteria:

* Active autoimmune disease requiring systemic immunosuppressive therapy
* Patients requiring corticosteroid therapy at a dose of 15 mg or more per day of prednisone or the equivalent; pulsed corticosteroid dose for disease control is acceptable
* Patients who are human immunodeficiency virus (HIV) seropositive
* Women who are pregnant or breastfeeding
* Significant cardiovascular diseases within the past 6 months including uncontrolled congestive heart failure (\> New York Heart Association \[NYHA\] class II), myocardial infarction, unstable angina, or uncontrolled arrhythmia
* History of severe immediate hypersensitivity reaction to cyclophosphamide or fludarabine
* History or presence of clinically relevant non-lymphoma central nervous system pathology, including seizures that are uncontrolled on anticonvulsant therapy (\>= 1 seizure in the last year), paresis, aphasia, stroke, severe brain injuries, dementia, Parkinson disease, cerebellar disease, or psychosis
* Treatment with any investigational agent on a different clinical trial within 4 weeks prior to enrollment, unless the patient is documented to be unresponsive to the therapy and at least 3 half-lives have elapsed prior to enrollment
* Treatment with any anti-CD19 or anti-CD20 antibody or antibody-drug conjugate therapy within 4 weeks before enrollment
* Previous treatment with CD19-targeted CAR T cells that has resulted in ongoing B cell aplasia at the time of enrollment; patients that demonstrate recovery of normal B cells (\>= 20 B cells/ul) by flow cytometry at any point 28 days or later after CD19 CAR T cell infusion will be considered to have functional loss of CD19 CAR T cells and are potentially eligible
* Patients with systemic disease without radiologic evidence of CNS involvement and with isolated CSF involvement detectable by flow cytometry are eligible for enrollment in systemic disease cohorts if neurologically asymptomatic. CSF involvement is not an exclusion for patients enrolled as a primary or secondary CNS lymphoma.
* Presence of active acute or chronic graft versus host disease (GVHD)
* Uncontrolled active infection (bacterial, fungal, viral, mycobacterial) not responding to treatment with intravenous antibiotics, antiviral or antifungal agents
* Patients with concurrent known additional malignancy that is progressing and/or requires active treatment; exceptions include squamous or basal cell carcinoma of the skin and low grade prostate carcinoma (Gleason grade =\< 6). Maintenance anti-hormone therapies for breast or prostate cancers are allowed and are not considered active treatment
* Patients with blood or platelet transfusion within 1 week prior to signing Consent A, or with platelets \< 50,000/mm\^3, neutrophils \< 750/mm\^3, or hemoglobin \< 8.5 g/dL, unless the cytopenias are considered by the treating physician to be largely due to marrow involvement by lymphoma