Mountains and other high places become even more annoying when it starts to rain. It rains a lot in this game, and when it rains, you can’t climb. You can try, but Link slips so many times you’ll almost inevitably run out of stamina before you find a stable ledge to stand on. It’s one of those Nintendo things, where a silly little detail was implemented simply to have more silly little details regardless of how it actually affected gameplay.
If you were already halfway up a mountain before it started raining, you’re buggered. You’ll either need to hope you landed somewhere sheltered so you can start a fire (in another awkward bit of menu management and item holding), or abandon all your progress and paraglide to an inn or somewhere else that passes the time.
If none of those options seem viable, you can always wait. Just wait, wait, wait until the game decides you can play it again. These are the sorts of annoyances Breath of the Wild is full of. Minor inconveniences and shows of disrespect toward the player’s time.