Am I liable for her, and her dog's injuries if I own my property and residence?

There is an adjacent residence that is normally accessed by foot and vehicle via a driveway that is on my private property. This has been the case for 40+ years. There is not a legal easement recorded for this driveway. Just a gentleman's agreement between the former adjacent property owner and my husband. Recently, a realtor used the driveway to enter the two properties and hand out flyers. She placed a flyer in my USPS mailbox (illegal?). She had a brief conversation with the resident of the other property. She had her small dog with her on a leash. She was not invited and the property is posted private/no trespassing. While she was walking on the driveway to leave, my dog left my house via a pet door, ran to the driveway, and attacked her dog. During the 'scuffle' she fell down and got some scrapes. No emergency medical service was requested or called. She left on her own with her dog. Her dog was treated by a vet and released (so she says). She initially lied and said my dog bit her reporting it to animal control. My sister was a witness chasing the dog out of the house and my dog did not bite her. Now she is asking me to pay for her vet expenses, chiropractic treatments, miscellaneous medical supplies and other stuff. She was on the property without permission and it's posted no trespassing. She had to cross another person’s private property, also posted, to even access the 'driveway' and bypassed a locked vehicle gate, though she did so on foot by just walking around it, leaving her car on the other side. The other property does have frontage on a public road, it's just never been used.