Am I liable for a refund issued by mistake?

Fair Debt Collection Practices Act **does not ** apply to creditors collecting their own debts. You might have similar rights under state law, though.

Every U.S. jurisdiction in which I have researched the issue recognizes a cause of action for money paid by mistake. Check out Form 7 in the Federal Rules of Civil Procedure for example http://www4.law.cornell.edu/rules/frcp/Forms.pdf

This search reveals many copies of the same form in various state rules: complaint for money paid by mistake - Google Search

Sorry, I misunderstood, I guess. They sent it to an outside collection agency then?

Incidentally, the Fair Credit and Billing Act, which is easy to Google, covers original creditors and their debt collection practices.
Also reference the Fair Credit Reporting Act if you like.