Scalable quantum networking requires quantum systems with quantum processing capabilities. Solid state spin systems with reliable spin-optical interfaces are a leading hardware in this regard. However, available systems suffer from large electron-phonon interaction or fast spin dephasing. Here, we demonstrate that the negatively charged silicon-vacancy centre in silicon carbide is immune to both drawbacks. Thanks to its 4A2 symmetry in ground and excited states, optical resonances are stable with near-Fourier-transform-limited linewidths, allowing exploitation of the spin selectivity of the optical transitions. In combination with millisecond-long spin coherence times originating from the high-purity crystal, we demonstrate high-fidelity optical initialization and coherent spin control, which we exploit to show coherent coupling to single nuclear spins with ∼1 kHz resolution. The summary of our findings makes this defect a prime candidate for realising memory-assisted quantum network applications using semiconductor-based spin-to-photon interfaces and coherently coupled nuclear spins.