A ball of clay falls freely to the hard floor. It does not bounce noticeably and it very quickly comes to rest. What, then, has happened to the energy the ball had while it was falling?

The energy is internal energy. Energy is never “used up”. The ball finally has no elevation and no compression, so the ball-Earth system has no potential energy. There is no stove, so no energy is put in by heat. The amount of energy transferred away by sound is minuscule.