It depends on what the damages to the property are. If your pet causes damages to the property and you are required to pay for those damages, then yes, you can keep your pet if you do so. However, if your landlord is asking you to pay for damages to the property and refuses to allow you to keep your pet, they cannot legally evict you based on the fact that you have a pet. In other words, if you agree to pay for any damages caused by your pet, you are allowed to keep it.