The problem of the 'hole-boring' (HB)-type of radiation pressure acceleration of ions by circularly polarized laser pulses interacting with overdense plasmas is considered in the regime where the dimensionless scaling parameter I/ρc3 becomes large. In this regime a non-relativistic treatment of the 'HB' problem is no longer adequate. A new set of fully relativistic formulae for the mean ion energy and 'HB' velocity is derived and validated against one-dimensional particle-in-cell simulations. It is also found that the finite acceleration time of the ions results in large energy spreads in the accelerated ion beam even under the highly idealized conditions of constant laser intensity and uniform mass density.