My quick thoughts are:
1. Keep climbing
When I first started climbing years ago I always got pumped. Nowadays I can end a session after climbing much harder but not be pumped. Answer: keep climbing, your stamina will grow.
2. Warm up. I find if I start the hard climbs too soon I get pumped too early. Stick to easy ones till you’re warmed up.
3. While you’re climbing don’t hold on tighter than you need to. Otherwise you’re wasting energy and getting pumped sooner.
4. Shake out regularly. Clench & unclench your fists and shake you hands and forearms. This helps clear the lactic acid from your forearms.
5. Use your legs as much as possible. If you’re on a climb where you can use your legs to take more of your weight then do so – take some weight off your arms.
6. If not completely sure of this one -but I’ve noticed exercising outside of climbing helps with my climbing strength and stamina. Makes sense I suppose.
Hope that helps.
Chris