A Study to Evaluate the Safety and Preliminary Efficacy of ATA3219 in Participants With Relapsed/Refractory B-cell Non-Hodgkin Lymphoma

Inclusion Criteria: * Histologically confirmed, R/R, B-cell NHL according to the 2022 revision of the World Health Organization classification of lymphoid neoplasms \[Alaggio 2022\] defined as any of the following: 1. LBCL 2. FL Grade 3b 3. MCL * The following criteria apply for details of prior treatment/therapy: R/R to at least 2 lines of therapy; if the most recent line of therapy was autologous hematologic cell transplant (HCT), relapse within 12 months of the transplant. * Measurable disease by scan (diagnostic positron emission tomography-positive and/or computed tomography-measurable) as per Lugano Classification \[Cheson 2014\]. Magnetic resonance imaging may be used when computed tomography with contrast is contraindicated or when mandated by local practice. * If sufficient archival material is not available from the latest relapse, a new tumor biopsy is required any time during screening, prior to conditioning chemotherapy. * Participants who have received prior CD19-directed therapy as the prior line of therapy: 1. must have achieved either a CR or partial response as a best response and maintained the response for ≥ 3 months after receiving CD19-directed treatment, and 2. must still have CD19+ disease as determined by a local laboratory. * Eastern Cooperative Oncology Group performance status ≤ 2 * Adequate organ function * Written informed consent as per protocol. * Participants are able to commit to the inpatient portion of the study, encompassing conditioning (if per the institution's standard practice), and frequent monitoring during Days 1-15, as well as remain within 1 hour travel time of the clinical site for 28 days after each infusion. Exclusion Criteria: * History of a human immunodeficiency virus infection or acute or chronic active hepatitis B or C infection. * History or presence of clinically relevant central nervous system (CNS) pathology. * Unresolved Grade 1-2 Immune effector cell-associated neurotoxicity syndrome (ICANS) or experienced Grade 3-4 ICANS from prior chimeric antigen receptor T-cell. * Unresolved graft-versus-host disease (GvHD) or Grade 3-4 acute GvHD from any prior therapy or moderate to severe chronic GvHD from any prior therapy. * History of any one of the following cardiovascular conditions: class III or IV heart failure as defined by the New York Heart Association \[The Criteria Committee of the New York Heart Association 1994\], cardiac angioplasty or stenting, myocardial infarction, unstable angina, or other clinically meaningful cardiac disease, within the past 6 months of study informed consent. * History of malignancies, other than R/R NHL, unless the participant has been disease-free for ≥ 1 year (certain noninvasive malignancies are allowed). * Active primary, CNS-only, or systemic plus CNS involvement by lymphoma, unless the CNS involvement has been effectively treated. * Active autoimmune disorders or inflammatory conditions that require systemic immunosuppressive therapies, including therapeutic doses of steroids. * Has received prior allogeneic HCT or prior solid organ transplant. * Systemic bacterial, viral, fungal, or other infection that is untreated or unresponsive to appropriate treatment (or requires IV antibiotics at enrollment); participants must be afebrile for ≥ 48 hours. Prophylactic antibiotics, antivirals, and antifungals are permitted. * Concurrent serious uncontrolled or unresolved medical condition, including any laboratory abnormality or psychiatric illness. * The following therapies within defined periods prior to the conditioning regimen: therapeutic doses of corticosteroids (\> 0.5 mg/kg/day of prednisone or equivalent), lymphodepleting chemotherapeutic agents, live attenuated vaccines, prior systemic cancer therapy, investigational agents, including approved drugs being used off label, autologous HCT, donor lymphocyte infusions, radiation, alemtuzumab. * Female who is breastfeeding or pregnant. * Inability or unwillingness to comply with study procedures. * Unwilling to use protocol specified contraceptive methods. * Life expectancy of ≤ 8 weeks. * For participants being considered for retreatment: had a DLT with prior ATA3219 dose.