I now have corroboration. My mom e-mailed me the details about dear Great Aunt Ollie, who was married to my maternal grandmother's brother. Here's what she had to say:
Yes, your great Aunt Ollie Sauls (married to Maw-Maw’s brother, Norman—who had alligators) had a pet squirrel named Peanut. She kept it in her house in a bird cage, but Peanut ran around in the house wherever and whenever he wanted most of the time.
Aunt Ollie was a chain smoker, a little eccentric and loved that squirrel. Peanut would eat out of her hand. I don’t remember how Peanut died, or even how Peanut came to be her pet, but I know she was crazy about him. I was always afraid of him.
I picture Peanut, the devious squirrel, staring down my mom as a little girl, a malicious glint in his eye. Probably daring her to make one move...
I never knew Aunt Ollie, but I did meet Norman a couple of times when I was little. He kept alligators in a trough on his property. I'm not sure what's up with that, except that my mom's mom's side of the family was always considered the "artsy" side (genteel Southern code for strange and eccentric).
It's generally accepted that my mom, my sister and I all get our creative bents (Mom is a musician, sis is an artist, I write) from that side. My maternal grandmother was an artist, too. My great-grandmother (mom's mom's mom)...well, she did keep peacocks, which is less creepy than alligators and a lot nicer to look at.