ABSTRACT The cuspy central density profiles of cold dark matter (CDM) haloes make them highly resilient to disruption by tides. Self-interactions between dark matter particles, or the cycling of baryons, may result in the formation of a constant-density core that would make haloes more susceptible to tidal disruption. We use N-body simulations to study the evolution of Navarro–Frenk–White (NFW)-like ‘cored’ subhaloes in the tidal field of a massive host, and identify the criteria and time-scales for full disruption. Our results imply that the survival of Milky Way satellites places constraints on the sizes of dark matter cores. We find that no subhaloes with cores larger than 1 per cent of their initial NFW scale radius can survive for a Hubble time on orbits with pericentres ${\lesssim} 10\, \mathrm{kpc}$. A satellite like Tucana 3, with pericentre ${\sim} 3.5\, \mathrm{kpc}$, must have a core size smaller than ${\sim} 2\, \mathrm{pc}$ to survive just three orbital periods on its current orbit. The core sizes expected in self-interacting dark matter models with a velocity-independent cross-section of $1\, \mathrm{cm^2}\,\mathrm{g}^{-1}$ seem incompatible with ultrafaint satellites with small pericentric radii, such as Tuc 3, Seg 1, Seg 2, Ret 2, Tri 2, and Wil 1, as these should have fully disrupted if accreted on to the Milky Way ${\gtrsim} 10\, \mathrm{Gyr}$ ago. These results suggest that many satellites have vanishingly small core sizes, consistent with CDM cusps. The discovery of further Milky Way satellites on orbits with small pericentric radii would strengthen these conclusions and allow for stricter upper limits on the core sizes.