We prove a Cell Decomposition Theorem for Henselian valued fields with analytic structure in an analytic Denef-Pas language. 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 analytic motivic integration and analytic motivic constructible functions in the line of R. Cluckers and F. Loeser [Fonctions constructible et int\'egration motivic I, Comptes rendus de l'Acad\'emie des Sciences, 339 (2004) 411 - 416].
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