Modified Immune Cells (AFM13-NK) and A Monoclonal Antibody (AFM13) in Treating Patients With Recurrent or Refractory CD30 Positive Hodgkin or Non-Hodgkin Lymphomas

Inclusion Criteria: * Patients with a diagnosis of relapsed or refractory classical Hodgkin lymphoma (HL), anaplastic large cell lymphoma (ALCL), peripheral T-cell lymphoma not otherwise specified (PTCL-NOS), mycosis fungoides (MF), or B-cell non-Hodgkin lymphoma with a pre-enrollment tumor biopsy positive for CD30 by immunohistochemistry at \>= 1%. Patients with HL, ALCL and MF must be refractory or intolerant to brentuximab vedotin. * Karnofsky performance status \>= 60%. * Absolute neutrophil count \>= 500/mm\^3 * Platelet count \>= 50,000/mm\^3 * Serum creatinine clearance \>= 50 ml/min, estimated using the Cockcroft-Gault equation. * Alanine aminotransferase (ALT) and/or aspartate aminotransferase (AST) =\< 3 x upper limit of normal (ULN). * Bilirubin =\< 2 x ULN. * Alkaline phosphatase (ALP) =\< 2 x ULN. * Forced expiratory volume in 1 second (FEV1) \>= 50% * Forced vital capacity (FVC) \>= 50% * Carbon monoxide diffusing capability test (DLCO) (corrected for hemoglobin \[Hgb\]) \>= 50% * Left ventricular ejection fraction \>= 40%. * No uncontrolled arrhythmias or symptomatic cardiac disease. * If female of child-bearing potential, must not be pregnant or be breastfeeding and required to have a negative urine or serum pregnancy test within 3 days prior to the first dose of study drug. * Note: Urine pregnancy tests that cannot be confirmed as negative, require a confirmatory negative serum pregnancy test. In addition, females of childbearing potential must agree use of a highly effective method of contraception for the course of the study from 14 days prior to the first dose of study drug until 60 days after the last dose of study drug. Non-childbearing potential is defined as: Postmenopausal: defined as no menses for 12 months without an alternative medical cause. A high follicle-stimulating hormone (FSH) level in the postmenopausal range may be used to confirm post-menopausal state in women not using hormonal contraception or hormonal replacement therapy. In the absence of 12 months of amenorrhea, FSH measurements indicating post-menopausal status must be documented in patient's medical history. Permanently sterile: documented permanent sterilization e.g. hysterectomy, bilateral salpingectomy and bilateral oophorectomy. If male, surgically sterile or agrees to use a highly effective method of contraception, 14 days prior to the first dose of study drug until 60 days after the last dose of study drug. Exclusion Criteria: * Major surgery \< 4 weeks prior to first dose of study drug. * Any other severe or uncontrolled disease or condition which might increase the risk associated with study participation. * Any other malignancy known to be active, with the exception of treated cervical intra-epithelial neoplasia and non-melanoma skin cancer. * Grade \>= 3 non-hematologic toxicity from prior therapy that has not resolved to grade =\< 2. * Active hepatitis B, either active carrier (hepatitis B surface antigen positive \[HBsAg +\]) or viremic (hepatitis B virus \[HBV\] deoxyribonucleic acid \[DNA\] \>= 10,000 copies/mL, or \>= 2,000 IU/mL), or hepatitis C (detectable viral load by hepatitis C virus \[HCV\] ribonucleic acid \[RNA\] polymerase chain reaction \[PCR\]). * Active infection requiring parenteral antibiotics. * Human immunodeficiency virus (HIV) infection. * Treatment within prior 2 weeks with any anti-cancer agent, investigational or approved. * Active central nervous system (CNS) involvement (untreated parenchymal brain metastasis or positive cytology of cerebrospinal fluid). * Life expectancy =\< 6 months. * Previous treatment with AFM13.