Dynamic compressive behaviour of basalt–polypropylene fibre-reinforced concrete (BPFRC) was experimentally investigated using a 75-mm-diameter split-Hopkinson pressure bar. The results showed that the addition of basalt fibre (BF) and polypropylene fibre (PF) is effective at improving the impact-resistance behaviour of concrete. The dynamic compressive strength, critical strain, and energy absorption capacity of BPFRC increased with increasing strain rate. At strain rates of 20–140 s−1, the addition of BF and PF significantly increased the dynamic compressive strength, critical strain, and energy absorption capacity of concrete. The dynamic increase factor of BPFRC increased linearly with the decimal logarithm of strain rate. The hybrid addition of BF and PF significantly improved the strain rate effect of the dynamic compressive strength. The strengthening and toughening mechanisms of BF and PF are discussed in detail. The proposed dynamic damage constitutive model can be used to accurately describe the dynamic stress–strain relationship of BPFRC.