“I’ll just have a slither.”
If somebody offers you a piece of cake and you want a thin slice, you can ask them to give you a sliver. A sliver is a small, thin piece of something cut or split off a larger piece. Not a slither. Slither is what a snake does when it moves smoothly over a surface with a twisting or oscillating motion.
Some dictionaries now accept slither as an alternative word for sliver, so I suppose you can say it if you really want to. But it’s probably best not to. It’s based on a mishearing and not really correct.
You can read about the correct use of sliver and slither and a whole lot more, in the Clifford & Co editorial style guide.