First, a liability-only insurance policy does not cover repairs to your vehicle. Liability insurance is required by law and pays when the policyholder is legally responsible for another person' crash-related damage. A vehicle owner with liabilty-only coverage pays out-of-pocket for all vehicle repairs.
Your policy must have physical damage protection to share deer-related repair expenses with an insurance company. Physical damage is divided into two separate coverages - comprehensive and collision. Both typically have deductibles. A deer accident is covered as a comprehensive claim if there is evidence of direct contact with the deer. Without this, the crash is a collision loss.
Policyholder responsibilities following a deer-hit or other physical damage loss: