If that’s corrosion, it’s not coming from inside the watch. The button would have bulged out from the case (if not popped off outright) were it from the only thing (the battery) which could cause it. Not to mention — the watch itself would have long since stopped functioning.
Looks to me much more like something is coming off your hands when you press the button and accumulating there, trapping the occasional bit of dust and sand. Lotion?
As for cleaning it … I’d rinse the watch under running warm tap water, then use some non-abrasive dish soap (like Dawn or Seventh Generation) with a soft toothbrush. If that doesn’t work, I’d consider letting the watch soak in a glass of water for fifteen minutes or so before trying again.
If even that doesn’t work … I’d consider it analogous to getting superglue or hot asphalt on the watch. It’s probably a lost cause at that point … so give dental picks a try but be ready to admit that it’s not something that’s going to get fixed (and that you may well break it in the attempt).
b&