Freezing would solve that, but ice crystals shred tissue on the way in and on the way back out. Engineers at the University of Minnesota tried vitrification, cooling organs into a glass-like state with no ice at all.
The team packed rat kidneys with cryoprotectant and iron oxide nanoparticles, vitrified them, and stored them for up to 100 days. To rewarm without cracking the tissue, they ran an alternating magnetic field through the organ, heating the nanoparticles inside the organ's blood vessels so it warmed rapidly and evenly.
After the nanoparticles were flushed out, the kidneys were transplanted into rats whose own kidneys had been removed, and the organs took over full renal function on their own. It is one successful model in one species, and scaling the same rewarming field evenly through an organ the size of a human kidney is the next hard problem.

Read the original study
Han et al., Nature Communications, 2023 · doi.org/10.1038/s41467-023-38824-8
A frozen kidney, fully alive again.
Source
- Han, Z., Rao, J. S., Gangwar, L., Namsrai, B.-E., Pasek-Allen, J. L., Etheridge, M. L., Wolf, S. M., Pruett, T. L., Bischof, J. C., & Finger, E. B. (2023). Vitrification and nanowarming enable long-term organ cryopreservation and life-sustaining kidney transplantation in a rat model. Nature Communications, 14(1), 3407. https://doi.org/10.1038/s41467-023-38824-8