Optoelectronic devices that provide non-classical light states on demand have a broad range of applications in quantum information science, including quantum-key-distribution systems, quantum lithography and quantum computing. Single-photon sources in particular have been demonstrated to outperform key distribution based on attenuated classical laser pulses. Implementations based on individual molecules, nitrogen vacancy centres or dopant atoms are rather inefficient owing to low emission rates, rapid saturation and the lack of mature cavity technology. Promising single-photon-source designs combine high-quality microcavities with quantum dots as active emitters. So far, the highest measured single-photon rates are ~ 200 kHz using etched micropillars. Here, we demonstrate a quantum-dot-based single-photon source with a measured single-photon emission rate of 4.0 MHz (31 MHz into the first lens, with an extraction efficiency of 38%) due to the suppression of exciton dark states. Furthermore, our microcavity design provides mechanical stability, and voltage-controlled tuning of the emitter/mode resonance and of the polarization state.