Banach spaces over fields other than $\mathbb{C}$?

Let me record my relatively superficial comments above as an answer. I think this might be of some use while we wait for Matt E to see and respond to the question.

Yes, the theory of Banach spaces over a complete ultrametric field has been extensively studied, for instance in the book by Schneider in Zev's answer. There are important applications to modern number theory, especially the theory of p-adic families of modular forms. As Mariano has alluded to above, one of the major players in this game happens to be Professor M. Emerton.

If you want good analogues of the theorems of Hahn-Banach etc., you need to do something to circumvent the fact that there is no non-Archimedean field which is both locally compact and algebraically closed (unlike $\mathbb{C}$ in the Archimedean case). One way to do this is by studying spherically complete fields: for these one has the Hahn-Banach theorem verbatim. For other fields some modifications are necessary (and possible).

See these course notes of Kiran Kedlaya for a nice introductory treatment of the material in the above paragraph.

[Having just found this question, and seeing my name being brought up a couple of times, I feel I should say something!]

For me, the most familiar context for functional analysis concepts such as normed or Banach spaces, other than the classical case of coefficients in $\mathbb R$ or $\mathbb C$, is when the coefficient field is a $p$-adic field, i.e. a finite extension of $\mathbb Q_p$. (More generally, one can consider spherically complete completions of $\mathbb Q_p$, but finite extensions are the most concrete examples of these, and are good enough for many applications in number theory.)
The book by Schneider that Zev mentioned is a basic reference for this theory, and the upshot is that all the standard theorems (Hahn--Banach, uniform boundedness, open mapping and closed graph, etc.) carry over. (Note also, in the definition of a norm that Zev quotes, that one asks for the ultrametric triangle inequality, which is stronger than the classical triangle inequality. This is natural when the coefficient field itself is non-archemedean, especially if you think about the basic examples: e.g. if $X$ is a compact topological space and you take your Banach space to be $\mathcal C(X,\mathbb Q_p)$ (the space of continuous $\mathbb Q_p$-valued functions on $X$), equipped with the sup norm, then this sup norm will indeed satisfy the ultrametric triangle inequality, since that is true of the $p$-adic absolute value itself.)

As Pete mentioned in his answer, I (and others) have used this theory a lot in our investigations of the $p$-adic properties of automorphic forms. In general, in thinking about the $p$-adic numbers, there are times when it helps to be very algebraic in ones psychology, thinking of the $\mathbb Z_p$ as a projective limit of $\mathbb Z/p^n$ and so on, but there are other times when it is more helpful to think in analytic terms, and then functional analytic tools and view-points can be invaluable.

I haven't read it, but the book Nonarchimedian Functional Analysis by Peter Schneider looks relevant. Over a non-archimedian field $K$, he defines a norm $q$ on a $K$-vector space $V$ to be a function $q:V\rightarrow\mathbb{R}$ for which

• $q(av)=|a|\cdot q(v)$ for any $a\in K$ and $v\in V$,
• $q(v+w)\leq\max\{q(v),q(w)\}$ for any $v,w\in V$, and
• $q(v)=0$ implies $v=0$.