Color centers in diamond are versatile solid state atomic-like systems suitable for quantum technological applications. In particular, the negatively charged silicon vacancy center (SiV) exhibits a narrow photoluminescence (PL) line and lifetime-limited linewidth in a bulk at cryogenic temperature. We present a low-temperature study of chemical vapour deposition (CVD)-grown diamond nano-pyramids containing SiV centers. The PL spectra feature a bulk-like zero-phonon line with ensembles of SiV centers, with a linewidth below 10 GHz which demonstrates very low crystal strain for such a nano-object.