You are quite correct: You should not be raising a credit note, because that would be creating a duplicate credit on your sales ledger account for the customer: One being the repayment, which offsets the overpayment, and the other being the credit note they're asking for. As a result, your accounts would be wrong - and worse, since you'd now have a credit on the ledger, it would show up on any statements you send to them, and they might deduct it from a future payment, thus compounding the problem.

What I suspect may be the problem is that the customer has messed up their purchase ledger (perhaps, for example, by entering one or more previous invoices twice, or entering another supplier's invoice(s) on your account - either of which could have led to them overpaying). As a result, they now have a mismatched balance on their own system - your repayment to them, which they have nothing to allocate it against.

If so, what they are effectively asking for is for *you* to "fix" their system, by providing them with something to match against the repayment.

Realistically, they need to look at their own accounts, work out what their cock-up is, and fix it themselves - *properly* rather than cutting corners and making matters worse by asking someone else to cock-up their own accounts so that theirs looks (but isn't) right.