Spin impurities in diamond can be versatile tools for a wide range of solid-state-based quantum technologies, but finding spin impurities that offer sufficient quality in both photonic and spin properties remains a challenge for this pursuit. The silicon-vacancy center has recently attracted much interest because of its spin-accessible optical transitions and the quality of its optical spectrum. Complementing these properties, spin coherence is essential for the suitability of this center as a spin-photon quantum interface. Here, we report all-optical generation of coherent superpositions of spin states in the ground state of a negatively charged silicon-vacancy center using coherent population trapping. Our measurements reveal a characteristic spin coherence time, T2*, exceeding 45 nanoseconds at 4 K. We further investigate the role of phonon-mediated coupling between orbital states as a source of irreversible decoherence. Our results indicate the feasibility of all-optical coherent control of silicon-vacancy spins using ultrafast laser pulses.