TITLE: $p$-adic exponentials for $p$-adic Lie groups
QUESTION [8 upvotes]: Let $G$ be a $p$-adic Lie group, $\text{Lie}(G)$ its Lie algebra.
Is there any reasonable notion of exponential map $\text{exp} : \text{Lie}(G)\to G$?

REPLY [5 votes]: 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.