If you can't promote them, will they leave?
If you can't promote them, find out how you can give them what they want without the promotion.
When you can’t promote someone who is asking, and if you don’t want to lose them, find out why they want the promotion: what they think the position will allow them to do that they can’t do now.
It’s usually possible to give someone more visibility or authority, to get them involved in decision-making, or even leading an initiative without giving them the promotion.
If all they want is more money or a different title then find out what they think THAT will give them and see how you can give it to them without the promotion.