I'm looking for good textbooks on analytic functions on Banach spaces over a non-Archimedean 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 Non-Archimedean 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 infinite-dimensional analysis seems never studied in the non-Archimedean setting. Already the p-adic analogs of analytic functions behave quite differently from their classical counterparts. General monographs on non-Archimedean analysis contain only minimal information on p-adic analytic functions. There are good introductions in the books by Robert and Koblitz. However there are some books devoted specifically to this subject:
Hu, Pei-Chu; Yang, Chung-Chun. Meromorphic functions over non-Archimedean fields. Dordrecht: Kluwer Academic Publishers. 2000;
A. Escassut, Analytic elements in $p$-adic analysis. Singapore: World Scientific, 1995.
Although they are a little older and Anatoly's remark on general monographs might apply here, maybe A. C. M. van Rooij: Non-archimedean functional analysis and W. H. Schikhof: Ultrametric Calculus still contain helpful information.