We present N-body simulations of a new class of self-interacting dark matter models, which do not violate any astrophysical constraints due to a non-power-law velocity dependence of the transfer cross-section which is motivated by a Yukawa-like new gauge boson interaction. Specifically, we focus on the formation of a Milky-Way-like dark matter halo taken from the Aquarius project and resimulate it for a couple of representative cases in the allowed parameter space of this new model. We find that for these cases, the main halo only develops a small core (∼1 kpc) followed by a density profile identical to that of the standard cold dark matter scenario outside of that radius. Neither the subhalo mass function nor the radial number density of subhaloes is altered in these models but there is a significant change in the inner density structure of subhaloes resulting in the formation of a large density core. As a consequence, the inner circular velocity profiles of the most massive subhaloes differ significantly from the cold dark matter predictions and we demonstrate that they are compatible with the observational data of the brightest Milky Way dwarf spheroidals (dSphs) in such a velocity-dependent self-interacting dark matter scenario. Specifically, and contrary to the cold dark matter case, there are no subhaloes that are more concentrated than what is inferred from the kinematics of the Milky Way dSphs. We conclude that these models offer an interesting alternative to the cold dark matter model that can reduce the recently reported tension between the brightest Milky Way satellites and the dense subhaloes found in cold dark matter simulations.