Single-electron pumps are set to revolutionize electrical metrology by enabling the ampere to be redefined in terms of the elementary charge of an electron(1). Pumps based on lithographically fixed tunnel barriers in mesoscopic metallic systems(2) and normal/superconducting hybrid turnstiles(3) can reach very small error rates, but only at megahertz pumping speeds that correspond to small currents of the order of picoamperes. Tunable barrier pumps in semiconductor structures are operated at gigahertz frequencies(1,4), but the theoretical treatment of the error rate is more complex and only approximate predictions are available(5). Here, we present a monolithic, fixed-barrier single-electron pump made entirely from graphene that performs at frequencies up to several gigahertz. Combined with the record-high accuracy of the quantum Hall effect(6) and proximity-induced Josephson junctions(7), quantized- current generation brings an all-graphene closure of the quantum metrological triangle within reach(8,9). Envisaged applications for graphene charge pumps outside quantum metrology include single-photon generation via electron-hole recombination in electrostatically doped bilayer graphene reservoirs(10), single Dirac fermion emission in relativistic electron quantum optics(11) and read-out of spin-based graphene qubits in quantum information processing(12).