I'm looking for good textbooks on analytic functions on Banach spaces over a nonArchimedean field. If you know one(s), please let me know.
One beautiful book is Peter Schneider's Nonarchimedean Functional Analysis, appeared in the Springer Monographs in Mathematics in 2006. A more analytic one (with less emphasis on Functional Analysis and more on Calculus) in Alain Robert's A course in $p$adic Analysis, GTM 198. But I still think the bible is S. Bosch, U. Güntzer and R. Remmert's NonArchimedean Analysis, appeared in the Grundlehren der mathematischen Wissenschaften, 261.
I am not sure why the question is about "analytic functions on Banach spaces". This subject from infinitedimensional analysis seems never studied in the nonArchimedean setting. Already the padic analogs of analytic functions behave quite differently from their classical counterparts. General monographs on nonArchimedean analysis contain only minimal information on padic analytic functions. There are good introductions in the books by Robert and Koblitz. However there are some books devoted specifically to this subject:
Hu, PeiChu; Yang, ChungChun. Meromorphic functions over nonArchimedean fields. Dordrecht: Kluwer Academic Publishers. 2000;
A. Escassut, Analytic elements in $p$adic analysis. Singapore: World Scientific, 1995.

$\begingroup$ There is a Bourbaki's volume of the resume of the theory of (possibly infinite dimensional) differential and analytic manifolds over Archimedean(i.e. real and complex) and nonArchimedean fields. It contains definitions and results but no proofs on the title subject. $\endgroup$ – Makoto Kato Oct 25 '12 at 19:01
Although they are a little older and Anatoly's remark on general monographs might apply here, maybe A. C. M. van Rooij: Nonarchimedean functional analysis and W. H. Schikhof: Ultrametric Calculus still contain helpful information.