The spatial resolution and fluorescence signal amplitude in stimulated emission depletion (STED) microscopy is limited by the photostability of available fluorophores. Here, we show that negatively-charged silicon vacancy (SiV) centers in diamond are promising fluorophores for STED microscopy, owing to their photostable, near-infrared emission and favorable photophysical properties. A home-built pulsed STED microscope was used to image shallow implanted SiV centers in bulk diamond at room temperature. The SiV stimulated emission cross section for 765-800 nm light is found to be (4.0 +/- 0.3) x 10^(-17) cm^2, which is approximately 2-4 times larger than that of the negatively-charged diamond nitrogen vacancy center and approaches that of commonly-used organic dye molecules. We performed STED microscopy on isolated SiV centers and observed a lateral full-width-at-half-maximum spot size of 89 +/- 2 nm, limited by the low available STED laser pulse energy (0.4 nJ). For a pulse energy of 5 nJ, the resolution is expected to be ~20 nm. We show that the present microscope can resolve SiV centers separated by <150 nm that cannot be resolved by confocal microscopy.