Abstract The unusually low velocity dispersion and large size of Crater II pose a challenge to our understanding of dwarf galaxies in the Lambda Cold Dark Matter (LCDM) cosmogony. The low velocity dispersion suggests either a dark halo mass much lower than the minimum expected from hydrogen cooling limit arguments, or one that is in the late stages of extreme tidal stripping. The tidal interpretation has been favoured in recent work and is supported by the small pericentric distances consistent with available kinematic estimates. We use N-body simulations to examine this interpretation in detail, assuming a Navarro-Frenk-White (NFW) profile for Crater II’s progenitor halo. Our main finding is that, although the low velocity dispersion can indeed result from the effect of tides, the large size of Crater II is inconsistent with this hypothesis. This is because galaxies stripped to match the observed velocity dispersion are also reduced to sizes much smaller than the observed half-light radius of Crater II. Unless its size has been substantially overestimated, reconciling this system with LCDM requires that either (i) it is not bound and near equilibrium (unlikely, given its crossing time is shorter than the time elapsed since pericentre), or that (ii) its progenitor halo deviates from the assumed NFW profile. The latter alternative may signal that baryons can affect the inner halo cusp even in extremely faint dwarfs or, more intriguingly, may signal effects associated with the intimate nature of the dark matter, such as finite self-interactions, or other such deviations from the canonical LCDM paradigm.