Chirped mirrors have underpinned advances in ultra-fast lasers based on bulk optics but have yet to be fully exploited in integrated photonics, where they could provide a means to engineer otherwise unattainable dispersion profiles for a range of nonlinear optical applications, including soliton frequency comb generation. The vast majority of integrated resonators for frequency combs make use of microring geometries, in which only waveguide width and height are varied to engineer dispersion. Here, we present an integrated photonic-crystal Fabry–Pérot resonator made of gallium phosphide (GaP), a material exhibiting a Kerr nonlinearity 200 times larger than that of silicon nitride and a high refractive index that permits the creation of strongly chirped photonic-crystal mirrors. Leveraging the additional degrees of freedom provided by integrated chirped mirrors, we disentangle optical losses from dispersion. We obtain an overall dispersion that is more anomalous than that achievable in both silicon nitride and gallium phosphide ring resonators with the same free-spectral range (FSR), while simultaneously obtaining higher quality factors than those of GaP ring resonators. With subharmonic pulsed pumping at an average power of 23.6 mW, we are able to access stable dissipative Kerr frequency combs in a device with a FSR of 55.9 GHz. We demonstrate soliton formation with a 3-dB bandwidth of 3.0 THz, corresponding to a pulse duration of 60 fs. This approach to cavity design based on photonic-crystal reflectors offers nearly arbitrary dispersion engineering over the optical transparency window of the nonlinear material.