This phase II trial studies the side effects of cord blood-derived expanded allogeneic natural killer cells (umbilical cord blood natural killer [NK] cells), rituximab, high-dose chemotherapy, and stem cell transplant in treating patients with B-cell non-Hodgkin's lymphoma that has come back (recurrent) or that does not respond to treatment (refractory). Immune system cells, such as cord blood-derived expanded allogeneic natural killer cells, are made by the body to attack foreign or cancerous cells. Immunotherapy with rituximab, may induce changes in body's immune system and may interfere with the ability of tumor cells to grow and spread. Drugs used in chemotherapy, such as carmustine, cytarabine, etoposide, lenalidomide, melphalan, and rituximab, work in different ways to stop the growth of cancer cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. A stem cell transplant using stem cells from the patient or a donor may be able to replace blood-forming cells that were destroyed by chemotherapy used to kill cancer cells. The donated stem cells may also replace the patient's immune cells and help destroy any remaining cancer cells. Giving cord blood-derived expanded allogeneic natural killer cells, rituximab, high-dose chemotherapy, and stem cell transplant may work better in treating patients with recurrent or refractory B-cell non-Hodgkin's lymphoma.