The realization of on-chip quantum interfaces between flying photons and solid-state spins is a key building block for quantum-information processors, enabling, e.g., distributed quantum computing, where remote quantum registers are interconnected by flying photons. Self-assembled quantum dots integrated into nanostructures are one of the most promising systems for such an endeavor thanks to their near-unity photon-emitter coupling and fast spontaneous emission rate. Here we demonstrate high-fidelity on-chip entanglement between an incoming photon and a stationary quantum-dot hole spin qubit. The entanglement is induced by sequential scattering of the time-bin encoded photon interleaved with active spin control within a microsecond, two orders of magnitude faster than those achieved in other solid-state platforms. Conditioning on the detection of a reflected photon renders the entanglement fidelity immune to the spectral wandering of the emitter. These results represent a major step towards realizing a quantum node capable of interchanging information with flying photons and on-chip quantum logic, as required for quantum networks and quantum repeaters.