Is a locally free sheaf of finite rank a projective object?

Question

Let $X$ be a locally ringed space, in the category of $\mathscr O_X$- modules, is a locally free sheaf of finite rank a projective object?

Answer 1

Let $X = \mathbb{P}^1$. Following this question, it is easy to show that the structure sheaf $\mathscr{O}$ is not projective. Indeed, Serre duality implies $\operatorname{Ext}^1(\mathscr{O}, \omega_{\mathbb{P}^1}) \cong H^0(\mathbb{P}^1, \mathscr{O})^\vee \neq 0$.

You can avoid Serre duality with a longer argument given here.