It is claimed [here](https://mathoverflow.net/a/148650/138661) 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. 



Proper scheme such that every vector bundle is trivial
According to [this paper](http://reh.math.uni-duesseldorf.de/~schroeer/publications_pdf/resolution.pdf) it is not known if every proper algebraic scheme admits nontrivial vector bundles. Partial results can be found [here](https://www.ams.org/journals/tran/2017-369-07/S0002-9947-2016-06813-0/S0002-9947-2016-06813-0.pdf).


