How do you handle a second negotiation after the budget’s already been stretched?

Had a situation recently where we put forward what was genuinely our final offer after stretching the budget as far as we could. The candidate came back the next day asking for a bit more due to a counter-offer elsewhere, but there’s no room to move on the number.

Interested in how others usually handle this stage, whether that’s how you frame the conversation, what you emphasise instead of salary, or how you draw a clear line while keeping things professional.