In number theory there are $p$-adic Banach spaces, and more generally there is the whole subject of $p$-adic functional analysis. Applications go back to work of Dwork (as explained systematically by Serre) in the 1960s on the Weil conjectures. For more recent work see [here][1] and [here][2], for example. There is a book on this subject by [Schneider][3].

[1]: http://www.math.uchicago.edu/~fcale/Files/Cole2.pdf
[2]: http://perso.ens-lyon.fr/laurent.berger/autrestextes/hangzhou.pdf
[3]: https://books.google.com/books/about/Nonarchimedean_Functional_Analysis.html?id=UDnwX-ng1qIC