I have just rescued a mouse from one of our cats.
Couldn't stand it as cat was batting it from one side of kitchen to the other.
I carried mouse to a far corner of the garden and put it down and instead of running away it tottered under a leaf and started washing its little mousey face. Then I noticed one of its legs was all distended and looked broken.
Couldn't bear to kill it. Feel bad to leave it to suffer though.
Dim cat is still looking for it under the sideboard.