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 kap