Another idea is to just memorize it. It's not an impossible task, and considering how important it is, I think nobody would complain about making some effort to memorize their keys.
That's how it should be, keeping the private key always with us under any circumstances. So besides me keeping it in a very safe place in some places, keeping it in your wallet is also a good option.
And I also plan to engrave my personal wallet on a small steel plate and wear it as a necklace because that way my personal wallet is always with me. And that's another option besides me storing it in a place that is safe from anything, and depending on the type of dangerous condition and I will anticipate if there are signs.
But I didn't think of carrying it with me or writing it on a piece of paper, because that would be risky too, as it's easy to damage. Memorizing it is a good option too, but make sure you never forget what you've memorized, because in difficult situations, in my experience it's very difficult to remember what you've memorized.