Abstract The shallow faint-end slope of the galaxy mass function is usually reproduced in Λ cold dark matter (ΛCDM) galaxy formation models by assuming that the fraction of baryons that turn into stars drops steeply with decreasing halo mass and essentially vanishes in haloes with maximum circular velocities Vmax < 20–30 km s−1. Dark-matter-dominated dwarfs should therefore have characteristic velocities of about that value, unless they are small enough to probe only the rising part of the halo circular velocity curve (i.e. half-mass radii, r1/2 ≪ 1 kpc). Many dwarfs have properties in disagreement with this prediction: they are large enough to probe their halo Vmax but their characteristic velocities are well below 20 km s−1. These ‘cold faint giants’ (an extreme example is the recently discovered Crater 2 Milky Way satellite) can only be reconciled with our ΛCDM models if they are the remnants of once massive objects heavily affected by tidal stripping. We examine this possibility using the APOSTLE cosmological hydrodynamical simulations of the Local Group. Assuming that low-velocity-dispersion satellites have been affected by stripping, we infer their progenitor masses, radii, and velocity dispersions, and find them in remarkable agreement with those of isolated dwarfs. Tidal stripping also explains the large scatter in the mass discrepancy–acceleration relation in the dwarf galaxy regime: tides remove preferentially dark matter from satellite galaxies, lowering their accelerations below the amin ∼ 10−11 m s−2 minimum expected for isolated dwarfs. In many cases, the resulting velocity dispersions are inconsistent with the predictions from Modified Newtonian Dynamics, a result that poses a possibly insurmountable challenge to that scenario.