Given a holomorphic vector bundle over a compact Riemann surface, we characterize when the Bergman space of holomorphic sections of the restriction of the bundle to an open set is infinite dimensional. This answers a problem raised by Sz{\H o}ke (2020), which was motivated by the results of Carleson and Wiegerinck for planar domains. This is joint work with Anne-Katrin Gallagher and Liz Vivas