I live in the country & my little tortie cat is a devil.
She only seems to be limited by the size of the cat flap as to what she brings in! Pigeons & Rabbits are no problem to her, along with the usual mice. Thankfully she's never fetched a rat in... so far
I keep a couple of boxes that cat food pouches come in, cut a small (mouse size) hole in one side seal them back up. The mice will hopefully run into them to take shelter. Then you can return them to the wild. NB take them a fair way away from your place if possible, as my cat has been known to recapture them from my garden.
I have repatriated many of the little chaps this way. Hope this is is of some help.