By a theorem of Grauert, every locally free sheaf $\mathscr{F}$ defined on a contractible subset $X$ of $\mathbb{C}^n$ is free.
Of course, if $\mathscr{F}$ is not locally free this is no longer true. For instance, take a closed analytic subvariety $Z \subset X$; then the ideal sheaf $\mathscr{I}_Z \subset \mathscr{O}_X$ is coherent but not locally free.