The next big step in organ transplantation will be custom grown personalized organs which don't trigger an immune response, and thus don't require immunosuppressants. This might be available in a limited form for some organs in our lifetimes.

However, I'm skeptical that this will extend maximum human lifespans. The transplant surgery itself is extremely risky for elderly people. Consider the risks of cognitive decline caused by general anesthesia, post-surgical infections, blood clots, sarcopenia caused by bed rest, etc.