Re: Enough is enough!
This might be a long shot, but is there any possibility that the child friend might be getting abused at home? Abuse doesn't necessarily have to be the kind that leaves bruises - it could be mental or verbal. What's the child's mother like (and whoever her significant other is, if she has one)?
Don't jump to conclusions just yet. It could also be the friend is lonely. I know too well how it feels when I was a kid and got the rare chance to hang with friends, and when I had to go home I was upset. Is your child this kid's only friend? Try talking to the child himself/herself and see if they give a reason for not wanting to go home. It sounds to me like the kid's mother is thrilled to have her kid out of her hair and that's probably why she's not doing much to corral him/her.
Even though you don't want to hurt the kid's feelings, you still need to lay down the law. Tell this mother that while you don't mind her child visiting, your house is not his/her house and they DO need to go home eventually. If Mommy can't be bothered to collect her child, then the child can't come to visit anymore.