It is true that over a contractible manifold all differentiable vector bundles are trivial. However the method of proof does not apply in the holomorphic category.
It is also true that a contractible one dimensional complex manifold has no non-trivial line bundles. (Shown using the exponential sequence and a bit of tinkering.)
Moreover, I have read that the topological and holomorphic classification of bundles are the same over Stein manifolds. (No adequate reference was given.)
Hence my question:
Is it true that all holomorphic vector bundles are trivial over a contractible complex manifold?
Probably this is false. In this case, could you give me counter-examples?