Here is a possible example that I have been speculating about, on and off, since around 2006: it might not fit the OP's question, and I may be talking rubbish, so I'd welcome corrections from the true connoisseurs (as opposed to this dilletant).

Call a short exact sequence of Banach spaces and bounded linear maps naively exact if it is exact as a s.e.s. of vector spaces (this is just to get round the fact that category-theoretic cokernels and epimorphisms in Ban don't behave as they do in Vect). Call a Banach space $E$ homologically flat if $E\widehat{\otimes}\underline{\quad}$ preserves the short naively exact sequences of Banach spaces, where the tensor is the projective tensor product a.k.a. the "right one from the POV of closed monoidal categories".

Now if I understand the literature correctly, the homologically flat Banach spaces are known: they are precisely the ${\mathcal L}^1$-spaces of Lindenstrauss and Pelczynski, which are roughly speaking those where each fin-dim subspace $E$ is contained in a fin-dim subspace $F$ which is uniformly isomorphic to $\ell_1^{|F|}$.

As $\ell_1^n$ is the "free Banach space" on an $n$-element set, this result could be airily if imprecisely captured by saying "the flat Banach spaces are those built locally from finitely generated free Banach spaces". But now this sounds awfully like the Govorov-Lazard theorem from ring theory: a module over a fixed commutative ring $R$ is flat iff it is a colimit of finitely generated free $R$-modules.

As far as I know the Banach-space result is proved using duality (duals of flat spaces are injective, and the injective Banach spaces admit a description as ${\mathcal L}^\infty$-spaces) and doesn't seem to have a proof along the lines of the GL-theorem. But one might hope to find a proof that made use of a meaningful analogy between the category of Banach spaces and bounded linear maps, and categories of modules...

(Of course the module-categories are abelian, while Ban is not, but there may be some way to exploit the "embedding" of Ban into an abelian envelope, cf. work of Noël, Waelbroeck and in greater generality Schneiders. Calling Theo Bühler... calling Theo Bühler...)