Study of Kappa Chimeric Antigen Receptor (CAR) T Lymphocytes Co-Expressing the Kappa and CD28 CARs for Relapsed/Refractory Kappa+ Non-Hodgkin Lymphoma and Chronic Lymphocytic Leukemia/Small Lymphocytic Lymphoma.

Note: During the period of cell procurement and CAR.κ.28 T cell production, subjects are allowed to receive additional standard of care chemotherapy to stabilize their disease if the treating physician feels it is in the subject's best interest. For subjects requiring bridging chemotherapy while awaiting manufacture of their CAR.κ.28 T-cells, details regarding treatment(s) administered including dose, frequency, number of cycles, etc. will be collected. Inclusion Criteria for the Study Unless otherwise noted, subjects must meet all of the following criteria to participate in this study: 1. Written informed consent and HIPAA authorization for release of personal health information. 2. Adults ≥18 years of age. 3. Diagnosis of relapsed/refractory chronic lymphocytic leukemia/small lymphocytic lymphoma OR histologically confirmed B-cell NHL, including the following types defined by WHO 2016: Aggressive Lymphomas: * DLBCL not otherwise specified (NOS) * T cell/histiocyte rich large B cell lymphoma; primary cutaneous DLBCL, leg type; EBV-positive DLBCL NOS; DLBCL associated with chronic inflammation; Lymphomatoid granulomatosis; Large B-cell lymphoma with IRF4 rearrangement; Intravascular large B-cell lymphoma; ALK-positive large B-cell lymphoma * Primary mediastinal (thymic) large B-cell lymphoma * High grade B-cell lymphoma with MYC and BCL2 and/or BCL6 rearrangement; high grade B-cell lymphoma, NOS * B-cell lymphoma, unclassifiable, with features intermediate between DLBCL and classical Hodgkin lymphoma * Transformation of indolent lymphoma or CLL to DLBCL will also be included * Burkitt lymphoma Indolent Lymphomas: * Follicular lymphoma grade 1-3b * Splenic marginal zone lymphoma * Extranodal marginal zone lymphoma of mucosa-associated lymphoid tissue * Nodal marginal zone lymphoma * Mantle cell lymphoma * Subjects with central nervous system (CNS) disease will not be excluded as long as it has been stable for 3 months Subjects with bone marrow only involvement are eligible 4. Subjects relapsed after autologous or allogeneic stem cell transplant are eligible for this study. 5. Subjects who have received prior CD19-directed CAR therapies for relapsed/refractory disease are eligible for this study. However, at least 3 months must have passed since the subject received CD19 CAR-T cells. 6. Patients with aggressive lymphomas must have relapsed or refractory disease after having received at least 2 prior lines of systemic therapy, including, at a minimum: * An anti-CD20 monoclonal antibody * An anthracycline containing chemotherapy regimen (if eligible) * An autologous stem cell transplant (if eligible) 7. For indolent lymphomas, subjects must have received at least 2 prior lines of therapy for their lymphoma 8. Subjects with specifically relapsed/refractory chronic lymphocytic leukemia/small lymphocytic lymphoma must have received at least 2 prior therapy regimens which can include, but not limited to: * A combination of an anti-CD20 monoclonal antibody and an alkylating agent, OR * A Bruton's Tyrosine Kinase Inhibitor, OR * A BCL-2 inhibitor in combination with an anti-CD20 monoclonal antibody 9. Subjects with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial at the investigator's discretion. 10. Kappa-positive expression on lymphoma or CLL/SLL tissue sample, or kappa restriction on flow cytometry (archival or fresh) as confirmed by institutional hematopathology standard (result must be confirmed at the time of cell procurement). 11. Karnofsky score of \> 60% 12. Female subjects of childbearing potential must be willing to use 2 methods of birth control or be surgically sterile, or abstain from heterosexual activity for the course of the study, and for 6 months after the study is concluded. Female subjects of childbearing potential are those who have not been surgically sterilized or have not been free from menses for \> 1 year. The two birth control methods can be composed of: two barrier methods or a barrier method plus a hormonal method to prevent pregnancy. Female subjects of childbearing potential will also be instructed to tell their male partners to use a condom. Exclusion Criteria for the Study Subjects meeting any of the following exclusion criteria will not be able to participate in this study (procurement, lymphodepletion and cell infusion): 1\. A history of intolerance to bendamustine or fludarabine. Note: subjects with known history of intolerance to bendamustine may be considered for lymphodepletion with cyclophosphamide and fludarabine at the discretion of the clinical investigator. 2 Subject is pregnant or lactating. 3 Current use of systemic corticosteroids at doses ≥10 mg prednisone daily or its equivalent; those receiving \<10 mg daily may be enrolled at discretion of investigator. 4 Active infection with HTLV, HCV (can be pending at the time of cell procurement; only those samples confirming lack of active infection will be used to generate transduced cells) defined as not being well controlled on therapy as well as no history of HIV. Subjects are required to have negative HIV antibody, negative HTLV1 and HTLV2 antibodies, negative hepatitis B surface antigen, and negative HCV antibody or viral load. Eligibility Criteria to be Met Prior to Procurement 1. Subject has signed a consent to undergo cell procurement. 2. Evidence of adequate organ function as defined by: * Total bilirubin \<1.5 × ULN (subjects with Gilbert's syndrome may be enrolled despite a total bilirubin level \>1.5 mg/dL if their conjugated bilirubin is \<1.5 × ULN) * AST and ALT \< 5x ULN * Pulse oximetry of \>90% on room air * Creatinine ≤ 2 x ULN 3. Imaging results from within 120 days prior to procurement to assess presence of active disease. 4. Confirmed kappa-positive expression on lymphoma or CLL/SLL tissue or bone marrow sample (archival or fresh) as confirmed by pathology. 5. Subject has adequate cardiac function, defined as: * No ECG evidence of acute ischemia * No ECG evidence of active, clinically significant conduction system abnormalities * Prior to study entry, any ECG abnormality at screening not felt to put the subject at risk has to be documented by the investigator as not medically significant * No uncontrolled angina or severe ventricular arrhythmia * Left ventricular ejection fraction (LVEF) \>40% as measured by ECHO, with no additional evidence of decompensated heart failure, performed within 30 days prior to procurement 6. In women of child-bearing potential, negative serum pregnancy test within 72 hours prior to procurement or documentation that the subject is post-menopausal. Post-menopausal status must be confirmed with documentation of absence of menses for \> 1 year. Eligibility Criteria to be Met Prior to Lymphodepletion 1. Written informed consent to enroll in the CAR-T cell therapy trial must be obtained prior to lymphodepletion. 2. The last bridging therapy should be completed at least 3 weeks prior to lymphodepletion. 3. Subjects who have received bridging therapy will be reassessed with imaging within 5 days prior to lymphodepletion and at least 3 weeks after bridging therapy. If a patient did not receive bridging chemotherapy, they will be imaged within 10 days prior to lymphodepletion. 4. Adequate organ function per the following criteria are required prior to lymphodepletion: * Adequate bone marrow function, as defined by: * ANC \>1.0 × 109/L * Platelets \>50 × 109/L unless related to lymphoma involvement (independent of transfusion within 7 days of lymphodepletion) * Total bilirubin ≤1.5 × ULN (subjects with Gilbert's syndrome may be enrolled despite a total bilirubin level \>1.5 mg/dL if their conjugated bilirubin is \<1.5× ULN) * AST and ALT ≤ 5× ULN * Pulse oximetry of \> 90% on room air * Creatinine ≤2 x ULN * If subjects display any clinical signs or symptoms of cardiac dysfunction after receiving bridging chemotherapy, they will undergo repeat ECG and ECHO to reassess their cardiac function and status 5. In female subjects of childbearing potential, a negative serum pregnancy test within 72 hours prior to l ymphodepletion or documentation that the subject is post-menopausal or has been surgically sterilized. Post-menopausal status must be confirmed with documentation of absence of menses for \> 1 year. 6. In subjects with CLL/SLL, a bone marrow biopsy within 28 days prior to lymphodepletion. 7. In subjects with WM/LPL, a bone marrow biopsy within 90 days prior to lymphodepletion. 8. Subjects must have autologous transduced activated T-cells that meet the Certificate of Analysis (CofA) acceptance criteria. 9. Has not received any tumor vaccines within the previous six weeks prior to lymphodepletion. 10. Has not received investigational agent or cancer-directed therapy within the previous 3 weeks, or 5 half-lives (whichever is shorter), prior to lymphodepletion. 11. Subjects may not be receiving strong inhibitors of CYP1A2 (e.g., fluvoxamine, ciprofloxacin) up through 72 hours after the last dose of bendamustine, as these may increase plasma concentrations of bendamustine, and decrease plasma concentrations of its metabolites. See http://medicine.iupui.edu/clinpharm/ddis/ for an updated list of strong inhibitors of CYP1A2 12. Subject is not taking a prohibited or contraindicated medication listed in the protocol. Contraindicated medications should be discontinued at least two weeks prior to the scheduled lymphodepletion or by at least 5 half-lives of the contraindicated medication, whichever is shorter. 13. No evidence of uncontrolled infection or sepsis. Eligibility Criteria to be Met Prior to Cell Infusion After Lymphodepletion 1. No evidence of uncontrolled infection or sepsis. 2. Evidence of adequate organ function as defined by: 1. Total bilirubin ≤2 × ULN, unless attributed to Gilbert's syndrome 2. AST \< 5 × ULN 3. ALT \< 5 × ULN 4. Creatinine ≤ 3 x ULN 3. Subject has no clinical indication of rapidly progressing disease in the opinion of the clinical investigator. 4. Subject is a good candidate for treatment with CAR.κ.28 cell product per the clinical investigator's discretion.