In reply to JStearn:
If you're not keen on the Olympus Em5 then I would also recommend the Sony RX100, specifically the RX100iii. For iii to v versions have the same lens, f1.8 - 2.8 and 24-70mm eq focal length, while the vi and vii have a longer zoom but slower aperture. In my opinion the brighter lens on the iii to v is more useful for climbing. The RX100iii is the sweet spot for cost imo, the improvements in the iv and v are mostly for video.
I have loads of photos from my RX100iii here: https://thomasogarden.wixsite.com/tomgarden/photography
It even does ok for astrophotography in a pinch, with the f1.8 lens.
Graham Zimmerman used an RX100 (not sure which version) for his Link Sar ascent. Pretty ringing endorsement. Take a look at some of his photos:
https://rockandice.com/climbing-news/link-sar-full-trip-report-on-first-asc...
My only caveat would be that the RX100 is expensive and fragile. Mine started to glitch out a lot, probably from moisture during repeated mountaineering trips. The lens cover also stopped working and the lens front element got scratched. I eventually replaced it with an Em5 because I liked the idea of not having to replace the whole camera when the lens gets damaged. I also have a Panasonic GM1 which I use for rock climbing. This is almost the same size as the RX100 with a small lens, and with a prime like the Panasonic 20mm f1.7 it can beat the RX100 in image quality.