  This paper is motivated by Grothendieck's splitting theorem. In the 1960s,
Gohberg generalized this to a class of Banach bundles. We consider a compact
complex manifold $X$ and a holomorphic Banach bundle $E \to X$ that is a
compact perturbation of a trivial bundle in a sense recently introduced by
Lempert. We prove that $E$ splits into the sum of a finite rank bundle and a
trivial bundle, provided $H^{1}(X, \O)=0$.
