If this was a negotiation then that might be the end. One of the great benefits of mediation is the presence of a third party who has been trained to assist parties through stalemate.
Mediators are trained to focus on the flow of the negotiation. If they sense that parties are becoming tired or stuck on a point, they may change the topic, or suggest a break. If the stalemate is extreme, they may suggest a caucus to speak privately with each side. In the joint session a mediator may ask an open question to stimulate further discussion.
The mediator may help by working with each party to generate options. The mediator may help a party consider what is possible for them even if it a creative solution not previously suggested.
Your mediator is your best resource; so make sure to use them.