By means of magnetization, specific heat, and muon-spin relaxation measurements, we investigate newly synthesized high-pressure oxidized Cu0.75Mo0.25Sr2YCu2O7.54, in which overdoping is achieved up to p similar to 0.46 hole/Cu, well beyond the T-c-p superconducting dome of cuprates, where Fermi-liquid behavior is expected. Surprisingly, we find bulk superconductivity with T-c = 84 K and superfluid density similar to those of optimally doped YBa2Cu3O7-delta. On the other hand, specific heat data display a large electronic contribution at low temperature, comparable to that of nonsuperconducting overdoped La2-x SrxCuO4. These results point at an unusual high-T-c phase with a large fraction of unpaired holes. Further experiments may assess the Fermi-liquid properties of the present phase, which would put into question the paradigm that the high T-c of cuprates originates from a non-Fermi-liquid ground state. ; Peer reviewed