Algebraic Geometry – Nonaffine Schemes and Split Exact Sequences of Vector Bundles

Question

Is there an example of a non-affine scheme $X$ such that every short exact sequence of vector bundles over $X$ splits? If there are such examples then what if we ask it to be true of all (not necessarily finite rank) locally free $\mathcal{O}_X$-modules

Answer 1

Affine plane with double origin works.