CAR.CD19-CD28-zeta-2A-iCasp9-IL15-Transduced Cord Blood NK Cells, High-Dose Chemotherapy, and Stem Cell Transplant in Treating Participants With B-cell Lymphoma

Inclusion Criteria: 1. Age 18-70. 2. Patients with B-cell lymphoma who are candidates to autologous stem-cell transplantation, including: 1. Primary refractory or relapsed diffuse large B-cell lymphoma in response to salvage treatment. 2. Primary refractory or relapsed follicular lymphoma or other indolent B-cell histology in response to salvage treatment. 3. Chemosensitive mantle-cell lymphoma in first or later line of treatment. 4. Patients with B cell lymphoma (all CD19+ NHL) with progressive or refractory disease who would otherwise not be candidates for autologous stem cell transplantation. 3. Adequate organ function: Renal: Creatinine clearance (as estimated by Cockcroft Gault) \>/= 60 cc/min. Hepatic: ALT/AST \</= 2.5 x ULN or \</= 5 x ULN if documented liver metastases, Total bilirubin \</= 1.5 mg/dL, except in subjects with Gilbert's Syndrome in whom total bilirubin must be \</= 3.0 mg/dL. Cardiac: Cardiac ejection fraction \>/= 50%, no evidence of pericardial effusion as determined by an ECHO, and no clinically significant ECG findings. Pulmonary: No clinically significant pleural effusion, Baseline oxygen saturation \> 92% on room air. 4. Patients must have a cord blood unit available which is matched with the patient at 4, 5, or 6/6 HLA class I (serological) and II (molecular) antigens. 5. Availability of autologous peripheral blood stem cell graft, containing at least 6.0 x 10\^6 CD34+ cells/kg. 6. Performance status \< 2 (ECOG). 7. Negative Beta HCG in woman with child-bearing potential. 8. All participants who are able to have children must practice effective birth control while on study. Acceptable forms of birth control for female patients include: hormonal birth control, intrauterine device, diaphragm with spermicide, condom with spermicide, or abstinence, for the length of the study. If the participant is a female and becomes pregnant or suspects pregnancy, she must immediately notify her doctor. If the participant becomes pregnant during this study, she will be taken off this study. Men who are able to have children must use effective birth control while on the study. If the male participant fathers a child or suspects that he has fathered a child while on the study, he must immediately notify his doctor. 9. Signed consent to long-term follow-up protocol PA17-0483. Exclusion Criteria: 1. Primary CNS lymphoma. 2. Grade \>/= 3 non-hematologic toxicity from prior therapy that has not resolved to \</= G1. 3. Prior whole brain irradiation. 4. Active hepatitis B, either active carrier (HBsAg +) or viremic (HBV DNA \>/= 10,000 copies/mL, or \>/= 2,000 IU/mL). 5. Evidence of either cirrhosis or stage 3-4 liver fibrosis in patients with chronic hepatitis C or positive hepatitis C serology. 6. Active infection requiring parenteral antibiotics. 7. HIV infection. 8. Radiation therapy in the month prior to enroll. 9. Presence of fungal, bacterial, viral, or other infection requiring IV antimicrobials for management. Note: Simple UTI and uncomplicated bacterial pharyngitis are permitted if responding to active treatment. 10. Concomitant use of other investigational agents.