Why does handling peeled chayote make your hands feel numb/rigid?
This problem can occur with several foods including squashes, pumpkins, zucchini, and even cucumber. From what I’ve read, it is caused by a sort of sap that the fruit emits when it is cut, and it’s worse in fruits that aren’t fully ripe. (Fruit is used in the botanical sense.)
Remove by  simply soak your hands in warm soapy water for about 5 minutes. Then scrub them with the scrubby side of a sponge. Be sure to scrub any surface area that had the film on it. You can soak your hands longer and less scrubbing will be required.