I've never completely understood what counts as "an application of category theory". With other areas of mathematics an "application" of area A to area B is generally a result which translates a problem in B into the language of A, solves the problem using the main theorems of A, and translates the solution back into the language of B. The problem is that I don't know what the main theorems of category theory are (or even if there are any "main theorems").
What I can say is that many interesting and nontrivial categories do arise in certain parts of functional analysis and it is useful to understand the structure of these categories. The specific part of functional analysis that I have in mind is the theory of operator algebras. For instance, in C*-algebra theory one considers a category whose objects are C*-algebras and whose morphisms are given by groups $KK(A,B)$ which simultaneously generalize K-theory and K-homology. Many of the deepest theorems in the subject are organized around the "Kasparov product" which is nothing more than the composition law $KK(A,B) \times KK(B,C) \to KK(A,C)$ in this category. KK-theory and its close cousin E-theory can be characterized according to homotopy invariance and various functorial properties.
On a related note, Connes' noncommutative geometry program (arguably part of functional analysis) relies heavily on the tools of category theory and homological algebra. Even at its inception the program was based on an analogy between de Rham cohomology and the periodic cyclic homology of a "smooth subalgebra" of a C*-algebra. In the process of investigating the relationship between cyclic homology and K-theory people have realized that it is useful to take seriously the category of projective modules over such smooth subalgebras rather than passing to K-theory. The work of Jonathan Block is particularly relevant; you might have a look here, for example.