Physical damage insurance is divided into comprehensive (also called other than collision) and collision coverages. Each has a deductible. A deer-hit is a comprehensive claim if the adjuster can verify direct contact with the animal. The damages are paid as a collision loss if you swerved to miss the deer and hit a tree, fence, or other fixed object.
Your insurance company cannot help you if you have a liability-only policy. Bodily injury and property damage are required by law, but pay for the other person's crash related injuries or property damage when you are legally responsible for a crash. Liability insurance does not pay for damage to your vehicle.
A comprehensive loss does not typically affect premium, but a collision claim often means higher rates. Specifics of the deer hit accident may cause the insurance company to do nothing; add an accident surcharge; terminate the policy; or take other action.