Usually even when shot (mortally but not yet dead) they go under and won't float up for a few hours. Some even dive down and bite on to or under limbs to stay down. Once in a while on a good shot they will flop around for a few seconds and then dive under.

What happens if you shoot a beaver?

Kill without trapping

Beavers are most active at dusk or dawn and best shot when they are out of the water. Using a small caliber gun or shooting a beaver while it is in the water will just end up in a slow dying, suffering beaver.

Do otters float when dead?

Otter are like beaver, they may float or sink if killed instantly with a head shot. If they don't die instantly they will sink.