Building upon the demonstration of coherent control and single-shot readout of the electron and nuclear spins of individual (31)P atoms in silicon, we present here a systematic experimental estimate of quantum gate fidelities using randomized benchmarking of 1-qubit gates in the Clifford group. We apply this analysis to the electron and the ionized (31)P nucleus of a single P donor in isotopically purified (28)Si. We find average gate fidelities of 99.95% for the electron and 99.99% for the nuclear spin. These values are above certain error correction thresholds and demonstrate the potential of donor-based quantum computing in silicon. By studying the influence of the shape and power of the control pulses, we find evidence that the present limitation to the gate fidelity is mostly related to the external hardware and not the intrinsic behaviour of the qubit.