A new ion radiation-pressure acceleration regime, the "leaky light sail," is proposed which uses sub-skin-depth nanometer foils irradiated by circularly polarized laser pulses. In the regime, the foil is partially transparent, continuously leaking electrons out along with the transmitted laser field. This feature can be exploited by a multispecies nanofoil configuration to stabilize the acceleration of the light ion component, supplementing the latter with an excess of electrons leaked from those associated with the heavy ions to avoid Coulomb explosion. It is shown by 2D particle-in-cell simulations that a monoenergetic proton beam with energy 18 MeV is produced by circularly polarized lasers at intensities of just 10¹⁹ W/cm². 100 MeV proton beams are obtained by increasing the intensities to 2 × 10²⁰ W/cm².