If you hit a wild animal
Hitting an animal on the road (the most common is a deer collision) is covered by comprehensive coverage. In some states, laws or insurance regulations keep car insurance companies from raising your rates if you make a comprehensive claim. Even in the states whose laws do not stop car insurance providers from raising rates due to a comprehensive claim, many auto insurance providers will not up your premiums due to a comprehensive claim. It is not the same for collision claims.
If you avoid hitting an animal and damage your car
If you are driving and swerve to avoid hitting animal, and as a result damage your car, such as driving into a ditch or a pole, you may file a claim under collision coverage. Without collision on your car insurance policy, you're out of luck. Collision claims often raise your rates, so you should evaluate the damage and decide if you'd rather pay out of pocket.
When a deer appears in front of you in the roadway it may be natural to try and dodge the animal but what various states' Department of Transportation offices advise is that if a deer jumps out in front of you and a collision seems inevitable, you should not attempt to swerve out of the way to avoid the animal. Swerving could cause you to lose control of your vehicle or place you in the path of an oncoming vehicle; either may end up causing more damage and injury then if you hit the deer.
If you injure your own pet
Technically, property damage liability insurance could pay veterinarian bills if you hit your own pet with your car. However, you could be found at fault for having allowed the pet to be in a dangerous area.
If you injure someone else's pet
You could very well using your car liability insurance to pay for someone else's veterinarian bill if you injure their pet with your car. You may have a case if the pet was off their leash in a driving area. In most states, you may not be 100 percent liable since the owner should have control over their pet.