The best answer I can provide is maybe. It really depends upon what damage is done by the animal and if it is your animal (i.e. pet) or a wild animal.
Your animal & physical damage to your home: more than likely not, unless your carrier offers an endorsement against owned pet damage. I personally have never heard of this type of coverage, but that doesn't mean it cannot be found in the surplus lines market.
Your animal & physical damage to another person or another home: sometimes yes. I say sometimes because if you have a breed of animal that is notorious for attacking people or causing damage your homeowners insurance will exclude it. However, if you have a breed that is typically docile and it randomly attacks a person or damages another person's property your liability coverage will take care of it.
Wild animal: more than likely not. If it is a rodent/pest then there is more than likely an exclusion in your policy. However, this exclusion might be something you could buy back (i.e. get coverage for) so check your policy. If it is something not considered a rodent/pest then it is very doubtful you have any coverage.
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Your animal & physical damage to your home: more than likely not, unless your carrier offers an endorsement against owned pet damage. I personally have never heard of this type of coverage, but that doesn't mean it cannot be found in the surplus lines market.
Your animal & physical damage to another person or another home: sometimes yes. I say sometimes because if you have a breed of animal that is notorious for attacking people or causing damage your homeowners insurance will exclude it. However, if you have a breed that is typically docile and it randomly attacks a person or damages another person's property your liability coverage will take care of it.
Wild animal: more than likely not. If it is a rodent/pest then there is more than likely an exclusion in your policy. However, this exclusion might be something you could buy back (i.e. get coverage for) so check your policy. If it is something not considered a rodent/pest then it is very doubtful you have any coverage.
I hope this helps.