In reply to berthathegerbil: It is possible to partially tear a tendon, your "specialist" is wrong.
If you hold, or get someone else to hold, all of your uninjured fingers still, then hold the middle knuckle of the injured finger still, can you still bend the tip? That's a really poor description of the examination, but it basically isolates your FDP tendon. If you can bend it that tendon is intact. Do the same for middle knuckle - if you can bend that with the rest isolated your FDS tendon is intact.
Sounds more likely that you've torn, or partially torn, a tendon pulley though. If it is you certainly shouldn't climb. Rest, ice and NSAIDs (ibuprofen) for the first 3-5 days. As the pain starts to subside introduce a GRADUAL rehab program that doesn't involve climbing. You could be looking at anywhere from 4 - 12 weeks recovery time. Loads of good articles online about pulley injuries and the how to rehab them.
Hope this has been of some help.