My cat was dropped, then listless and sleepy all day, body soft but can run and eat

Whether a fall is life-threatening depends on internal injury; internal damage usually cannot self-heal and untreated can be fatal. If the cat moves normally and eats and is alert, 3 days of rest usually brings recovery.

Cats hide pain well, so mild internal injury is hard to see. With lethargy or poor appetite after a fall, see a vet for internal bleeding, fractures or soft-tissue bruising to avoid delayed sequelae.