By the Syzygy theorem it follows that any coherent $\mathcal{O}_{\mathbb{P}^n}$-module $\mathcal{F}$ has a locally free resolution $$\dots \to F_1 \to F_0 \to \mathcal{F} \to 0,$$ where the $F_i$ are locally free $\mathcal{O}_{\mathbb{P}^n}$-modules.
If I understand correctly, it is not in general possible to construct locally free resolutions, and my question is simply:
Question: Are there any simple examples of a complex manifold $X$ and a coherent $\mathcal{O}_X$-module $\mathcal{F}$ which does not have a locally free resolution?
Hints, comments, and references are appreciated.