There are two questions here, in reality, I think.
First, in brief, I am told by many people that I "do functional analysis in the theory of automorphic forms", and I certainly do find a categorical viewpoint very useful. Second, in brief, it is my impression that the personality-types of many people who'd style themselves "(functional) analysts" might be hostile to or disinterested in the worldview of any part of (even "naive") category theory.
In more detail: as a hugely important example, I think the topology on test functions on $\mathbb R^n$ is incomprehensible without realizing that it is a (directed) colimit (direct limit). The archetype of incomprehensible/unmotivated "definition" (rather than categorical characterization) is in Rudin's (quite admirable in many ways, don't misunderstand me!) "Functional Analysis"' definition of that topology.
That is, respectfully disagreeing with some other answers, I do not think the specific-ness of concrete function spaces reduces the utility of a (naive-) categorical viewpoint.
From a sociological or psychological viewpoint, which I suspect is often dominant, it is not hard to understand that many people have a distaste for the structuralism of (even "naive", in the sense of "naive set theory") category theory. And, indeed, enthusiasm does often lead to excess. :)
I might claim that we are in a historical epoch in which the scandals of late 19th and early 20th century set theory prey on our minds (not to mention the mid-19th century scandals in analysis), while some still react to the arguable excesses of (the otherwise good impulses of) Bourbaki, react to certain exuberances of category theory advocates... and haven't yet reached the reasonable equilibrium that prosaically, calmly, recognizes the utilities of all these things.
Edit: since this question has resurfaced... in practical terms, as in L. Schwartz' Kernel Theorem in the simplest case of functions on products of circles, the strong topology on duals of Levi-Sobolev spaces is the colimit of Hilbert space topologies on duals (negatively-indexed Levi-Sobolev spaces) of (positively-indexed) Levi-Sobolev spaces. As I have remarked in quite a few other places, it was and is greatly reassuring to me that a "naive-categorical" viewpoint immediately shows that there is a unique (up to unique isomorphism) reasonable (!) topology there...
Similarly, for pseudo-differential operators, and other "modern" ideas, it is very useful to recast their description in "naive-categorical" terms, thereby seeing that the perhaps-seeming-whimsy in various "definitions" is not at all whimsical, but is inevitable.
A different example is characterization of "weak/Gelfand-Pettis" integrals: only "in my later years" have I appreciated the unicity of characterization, as opposed to "construction" (as in a Riemann/Bochner integral).