We performed a series of high-resolution N-body simulations to examine whether dark matter candidates in the form of primordial black holes (PBHs) can solve the cusp–core problem in low-mass dwarf galaxies. If some fraction of the dark matter in low-mass dwarf galaxies consists of PBHs and the rest is cold dark matter, dynamical heating of the cold dark matter by the PBHs induces a cusp-to-core transition in the total dark matter profile. The mechanism works for PBHs in the 25–100 M_⊙ mass window, consistent with the Laser Interferometer Gravitational-Wave Observatory (LIGO) detections, but requires a lower limit on the PBH mass fraction of 1 |${{\rm per\ cent}}$| of the total dwarf galaxy dark matter content. The cusp-to-core transition time-scale is between 1 and 8 Gyr. This time-scale is also a constant multiple of the relaxation time between cold dark matter particles and PBHs, which depends on the mass, the mass fraction, and the scale radius of the initial density profile of PBHs. We conclude that dark matter cores occur naturally in haloes composed of cold dark matter and PBHs, without the need to invoke baryonic processes.