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 <a href="https://www.sciencedirect.com/science/article/pii/0022404994901236">*here*</a>.    This suggests however that it's not easy to say much about the use of exponentials in $p$-adic groups.