The main results of this paper are a Cell Decomposition Theorem for Henselian
valued fields with analytic structure in an analytic Denef-Pas language, and
its application to analytic motivic integrals and analytic integrals over
\FF_q((t)) of big enough characteristic. To accomplish this, we introduce a
general framework for Henselian valued fields K with analytic structure, and
we investigate the structure of analytic functions in one variable, defined on
annuli over K. We also prove that, after parameterization, definable analytic
functions are given by terms. The results in this paper pave the way for a
theory of \emph{analytic} motivic integration and \emph{analytic} motivic
constructible functions in the line of R. Cluckers and F. Loeser
[\emph{Fonctions constructible et int\'egration motivic I}, Comptes rendus de
l'Acad\'emie des Sciences, {\bf 339} (2004) 411 - 416]
