A.C.
Be cordial to the mom, and nice to the child, but no more playdates or invitations over for the child.
Explain to your daughter that some people have problems with emotional balance, and that the best thing to do is to distance yourself from those people. That means no one-on-one stuff. Group activities are fine, but explain to your daughter that healthy, happy people do not act this way, and that if they cannot help themselves or don't know how to behave appropriately, then it's just better to avoid or minimize contact with them. But also explain that she should be nice, but that that doesn't mean your daughter needs to be friends (being nice/polite does not = friendship, and is perfectly okay). And as soon as school starts, start inviting over some of the kids in her class so she can start having other friends there.