$p$-adic exponentials for $p$-adic Lie groups

Besides the books already mentioned, I highly recommend Michel Lazard's Groupes analytiques p-adiques, which is the original source for a lot of the material in both Dixon-DuSautoy-Mann-Segal's Analytic pro-p groups and Schneider's p-adic Lie groups. Lazard's text was most probably written in close collaboration with Serre and is freely available online: http://www.numdam.org/item?id=PMIHES_1965__26__5_0

The exponential map and the Hausdorff formula are treated in particular (with a look towards $\mathbb{Z}_p$-integrality) in section 3.2.

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.