Suppose that $(M,\mathcal{O})$ is a $k$-ringed space, where $k$ is a field. By $k$-ringed I mean the structure sheaf is a sheaf of $k$-algebras with local stalks and residue field $k$. Suppose that $V$ is a sheaf of vector spaces over $k$ such that $V\otimes_k \mathcal{O}$ is locally free of finite rank. Does it follow that $V$ is a local system, i.e. locally constant?
The assumption implies that the stalks $V$ are of locally constant dimension.
I am interested in the case where $M$ is an analytic space/variety (so locally connected, hausdorff, etc. can be assumed).