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/Bourbaki 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.

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.