Currently I'm thinking you put a dropper over a faith plate which launches the sphere across the room. The sphere hits the button, and drops into a fizzler. Dropper makes new sphere, and it starts again. Still verifying game cache or I would test it myself.
EDIT: Fizzler isn't even needed. Just have the button trigger the dropper. It will spawn a new one when it gets pressed. You can use a fizzler as a NOT gate.
EDIT2: I made a proof of concept.
Note: Spoiler contains puzzle solution.