Let A be a Banach ring, and X/A be a projective scheme of finite type, endowed with a semi-ample line bundle L. Assuming that A is a geometric base ring, we define a class PSH(X,L) of plurisubharmonic metrics on L on the Berkovich analytification X^an and prove various basic properties thereof. We focus in particular on the case where A is a hybrid ring of complex power series and X/A is a smooth variety, so that X^an is the hybrid space associated to a degeneration X of complex varieties over the punctured disk. We then prove that when L is ample, any plurisubharmonic metric on L with logarithmic growth at zero admits a canonical plurisubharmonic extension to the hybrid space X^hyb . We also discuss the continuity of the family of Monge-Amp\`ere measures associated to a continuous plurisubharmonic hybrid metric. In the case where X is a degeneration of canonically polarized manifolds, we prove that the canonical psh extension is continuous on Xhyb and describe it explicitly in terms of the canonical model (in the sense of MMP) of the degeneration.