  Poincar\'e series of $p$-adic, definable equivalence relations have been
studied in various cases since Igusa's and Denef's work related to counting
solutions of polynomial equations modulo $p^n$ for prime $p$. General
semi-algebraic equivalence relations on local fields have been studied
uniformly in $p$ recently in \cite{16}. Here we generalize the rationality
result of \cite{16} to the analytic case, unifomly in $p$, building further on
the appendix of \cite{16} and on \cite{13b}, \cite{03}. In particular, the
results hold for large positive characteristic local fields. We also introduce
rational motivic constructible functions and their motivic integrals, as a tool
to prove our main results.
