A cat or a catamount fits this description...
a caterpillar
duck
Three letter animals that start with the letter c:catcodcow
Cher
carnivore
cradle
cat
· Cuckoo · Caiman
colt
a caterpillar
duck
Three letter animals that start with the letter c:catcodcow
Cher
carnivore
Cat, and a cow
"Cabbage".
The animal you're looking for is a "rabbit." Its front teeth, specifically the incisors, have a third letter "C" in the spelling of the word. Rabbits are known for their continuously growing teeth, which they need to gnaw on various materials to keep them trimmed.