Heloise, I had a great-aunt like this as a child. She was always asking for our stuff, and gave almost exactly the same speech you’ve described when we told her “No.”
Eventually, my mother got very tired of hearing her even ask. (We almost never gave her anything she asked for, but she kept it up all the same.) Thus, the following conversation was born:
Asky Aunt: “Oh, I just love your whatchamacallit! I could really use one just like it…”
Mom: “It is nice, isn’t it? We like it too.”
A.A.: “Since you have so many whatchamacallits, I wonder if you’d mind if I took this one?”
Mom: “I think we’d like to keep it ourselves.”
A.A.: “I really like it…”
Mom: “Thanks. So do we.”
A.A.: “I don’t ask for much… I’d really like to have it…”
Mom: “Actually, you ask for our things all the time. We tell you “no” because they are our things and we enjoy them. We’ve worked for all of our whatchamacallits and we want to keep them. We do not want to give them away.”
A.A.: “Well, that’s not very Christian of you!”
Mom: rolls eyes “I don’t care. We’re not giving you anything.”
Asky Aunt made like everything was just fine, then complained to some other relatives and family friends after her visit. They told her that it bothered them when she asked for things too. She got mad at everyone and stopped visiting for a while. Then she got lonely, and everything was okay again. She doesn’t ask for things anymore (Well, every once in a blue moon, but only needs to be reminded of above.) and the other family members thanked my mother for having the guts to say something.
Just my own experience. YMMV.