  We prove that for any FAb compact $p$-adic analytic group $G$, its
representation zeta function is a finite sum of terms
$n_{i}^{-s}f_{i}(p^{-s})$, where $n_{i}$ are natural numbers and
$f_{i}(t)\in\mathbb{Q}(t)$ are rational functions. Meromorphic continuation and
rationality of the abscissa of the zeta function follow as corollaries. If $G$
is moreover a pro-$p$ group, we prove that its representation zeta function is
rational in $p^{-s}$. These results were proved by Jaikin-Zapirain for $p>2$ or
for $G$ uniform and pro-$2$, respectively. We give a new proof which avoids the
Kirillov orbit method and works for all $p$. First part of arXiv:2007.10694,
second part uploaded as a separate paper.
