Paraffin wax! Your body temperature will soften it into malleable putty, but it hardens when you put it down, so lint and fur don't stick to it as much. I buy bulk spa-grade beads on amazon (scented options too!), form a little ball, and have at it. Each ball lasts a couple of weeks before getting too gross with hand oils, and then I toss it and make a fresh one.

Alternatively, those pop fidget toys are super satisfying for busy fingers and have amazing tactile feedback, though they aren't silent.