[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.