Hi there. Yes she does chase small animals. When we’re out in the woods she goes sniffing about and if she sees a rabbit she will chase it. I’ve also seen her chase birds in the garden. She’s never manages to catch anything but I somehow think she’ be quite interested in your pet rat. Really sorry about that. Deerhound dogs do make lovely pets but as you say being hounds they have a bit of a hunter in them. I wouldn’t chance it. Thanks for the interest anyway.Tanya