Does frostbite permanently discolor the skin?

It may discolor skin. Frostbite, like burns, comes in degrees. A simple first degree may not have any lingering effect, but the deeper degrees can cause a permanent discoloration of the skin due to underlying tissue necrosis. Not only is the dermis damaged, but also the nerves around blood vessels may be damaged, meaning that the vessels do not respond to temperature changes normally and that can cause color changes.
It can… Like any other deep skin injury, frostbite can permanently change skin color. Don't give up on some improvement until two years after the injury, however, as it can take some time. Massage with cocoa butter or the more expensive, but in my opinion, no more effective, scar fade products, to hasten the improvement.
Frostbite pigmenatio. Possibly. Any time skin is injured, it can affect the function of the melanocytes or cells responsible for pigmenting the skin. Speeding recovery with proper local wound care and avoiding sun exposure for the first year after the injury can lessen this effect, but it is still not completely preventable. Some of these effects may be treatable with tattooing or lasers in the future.