  The shallow faint-end slope of the galaxy mass function is usually reproduced
in $\Lambda$CDM galaxy formation models by assuming that the fraction of
baryons that turns into stars drops steeply with decreasing halo mass and
essentially vanishes in haloes with maximum circular velocities $V_{\rm
max}<20$-$30$ km/s. 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, $r_{1/2}\ll 1$ kpc). Many dwarfs have properties in disagreement with
this prediction: they are large enough to probe their halo $V_{\rm max}$ but
their characteristic velocities are well below $20$ km/s. These `cold faint
giants' (an extreme example is the recently discovered Crater 2 Milky Way
satellite) can only be reconciled with our $\Lambda$CDM 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 $a_{\rm min}\sim 10^{-11} m/s^2$ 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.
