It is claimed here that there exist proper schemes (probably over a field but not explicitly stated) with trivial Picard group. This means that every locally free $O_X$-module of rank 1 is trivial.

Do there exist proper schemes over a field such that every locally free $O_X$-module of finite rank is trivial?

Maybe it is easier to give some examples with algebraic spaces, but the accepted answer should give a scheme.

EDIT: examples should be positive-dimensional.