Hi and Welcome Erik!
Your story is absolutely charming and I hope it works out for the best for you both. One thing to keep in mind in the predator column: The list of animals willing to take on a chicken for fun or for food is disheartening: raccoons, skunks, possums, fox, coyote, raptors, dogs... So I hope you are able to catch her and follow the advice to shelter her in a more confined area until you two figure out your new "relationship." (Beware: Chickens can be ADDICTIVE!)
I started my flock just under a year ago and love these gals as pets as well as producers. Backyard Chickens is a terrific resource for information, comfort, consolation and friendship.
All the Best!