The dev doesn't seem to have added saves to the game yet. If you buy the carrot and then close and reopen the game, that upgrade will be lost again.
oh alright
No, it saves the progress correctly, but with a delay. The last few actions might be lost, just continue playing and it will save the progress. The previous progress is wiped in this version, but it will likely be compatible in future updates
Weird, I made it through all the upgrades and closed the game. When I relaunched it, it hadn't saved. However, it is now saving, so this is annoying.