We use N-body simulations to study the effects that a divergent (i.e. `cuspy') dark matter profile introduces on the tidal evolution of dwarf spheroidal galaxies (dSphs). Our models assume cosmologically motivated initial conditions where dSphs are dark-matter-dominated systems on eccentric orbits about a host galaxy composed of a dark halo and a baryonic disc. We find that the resilience of dSphs to tidal stripping is extremely sensitive to the cuspiness of the inner halo profile; whereas dwarfs with a cored profile can be easily destroyed by the disc component, those with cusps always retain a bound remnant, even after losing more than 99.99 per cent of the original mass. For a given halo profile, the evolution of the structural parameters as driven by tides is controlled solely by the total amount of mass lost. This information is used to construct a semi-analytic code that follows the tidal evolution of individual satellites as they fall into a more massive host, which allows us to simulate the hierarchical build-up of spiral galaxies assuming different halo profiles and disc masses. We find that tidal encounters with discs tend to decrease the average mass of satellite galaxies at all galactocentric radii. Of all satellites, those accreted before re-ionization (z >~ 6), which may be singled out by anomalous metallicity patterns, provide the strongest constraints on the inner profile of dark haloes. These galaxies move on orbits that penetrate the disc repeatedly and survive to the present day only if haloes have an inner density cusp. We show that the size-mass relationship established from Milky Way (MW) dwarfs strongly supports the presence of cusps in the majority of these systems, as cored models systematically underestimate the masses of the known ultra-faint dSphs. Our models also indicate that a massive M31 disc may explain why many of its dSphs with suitable kinematic data fall below the size-mass relationship derived from MW dSphs. We also examine whether our modelling can constrain the mass threshold below which star formation is suppressed in dark matter haloes. We find that luminous satellites must be accreted with masses above 108-109Msolar in order to explain the size-mass relation observed in MW dwarfs.