Let $G$ be a $p$-adic Lie group, $\text{Lie}(G)$ its Lie algebra.


Is there any reasonable notion of exponential map $\text{exp} : \text{Lie}(G)\to G$?



$p$-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.


