In number theory there are $p$-adic Banach spaces and $p$-adic Banach algebras (e.g., the Tate algebras), and more generally there is the whole subject of $p$-adic functional analysis. Applications of $p$-adic functional analysis go back to the 1960s in work of Dwork (as explained systematically by [Serre][1], go [here][2] for an English translation by Jay Pottharst) on the Weil conjectures. For more recent work see [here][3], [here][4], and [here][5]. There is a book on this subject by [Schneider][6].

[1]: https://eudml.org/doc/103829
[2]: http://vbrt.org/writings/serre-3.pdf
[3]: http://www.math.uchicago.edu/~fcale/Files/Cole2.pdf 
[4]: http://math.stanford.edu/~conrad/papers/aws.pdf
[5]: http://perso.ens-lyon.fr/laurent.berger/autrestextes/hangzhou.pdf
[6]: https://books.google.com/books/about/Nonarchimedean_Functional_Analysis.html?id=UDnwX-ng1qIC