My elderly neighbor has Alzheimers and "steals" my cat

Before she was impaired, I would allow my cat to spend the night with my neighbor after her dog died and she was lonely. Now I forbid it because she was hitting and kicking my cat. Told out-of-state person responsible for neighbor that cat is not allowed in neighbor's home; also reinforce with neighbor on a daily basis. She is leaving her door open to entice my cat in, and searching for my cat by entering my garage and looking on my property (back patio area). I have confronted her multiple times and she screams at me and says she can go where she wants. I learned that cat scratched neighbor on stomach when she tried to pick her up to put her in (or let her out) of her home. I am trying my best to keep cat away from her. It is not an option to keep her in my house because of another pet. Cat sleeps in enclosed patio in the evening and usually will nap in my garage or on the patio during the day. Out-of-state family is blaming me for the situation insisting it is my fault.