Serpents you find during the night or during the storm will despawn. The ones you find on a sunny day (this is rare) will not despawn.
Also, if you somehow manage to cage them, they will not despawn no matter when or where you found them. A friend and I have caged 3 of them, one found in a sunny day, one in a storm and one at night. We call all of them Steve and we feed them with boars for no particular reason :)
I guess workbenches around our custom made pool prevent them from despawning.

A tip for solo catching and killing a serpent on the land. Dig a hole and drag it in. It will keep him there while your stamina regenerates. Or use tamed wolves to do the dirty work for you. Set them to wait on shore and drag a serpent to them.