This is just a comment (in community-wiki format). I don't know how to cite an article efficiently otherwise.

As YCor points out, this notion has become fairly standard in the development of Lie groups over $p$-adic fields. On the other hand, there is not much explicit literature along these lines beyond Bourbaki. One paper that might be of interest is here. This suggests however that it's not easy to say much about the use of exponentials in $p$-adic groups.