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Aims. Doubly ionized silicon (Si III) is a powerful tracer of diffuse ionized gas inside and outside of galaxies. It can be observed in the local Universe in ultraviolet (UV) absorption against bright extragalactic background sources. We here present an extensive study of intervening Si III-selected absorbers and study the properties of the warm circumgalactic medium (CGM) around low-redshift (z <= 0.1) galaxies. Methods. We analyzed the UV absorption spectra of 303 extragalactic background sources, as obtained with the Cosmic Origins Spectrograph (COS) on-board the Hubble Space Telescope (HST). We developed a geometrical model for the absorption-cross section of the CGM around the local galaxy population and compared the observed Si III absorption statistics with predictions provided by the model. We also compared redshifts and positions of the absorbers with those of similar to 64 000 galaxies using archival galaxy-survey data to investigate the relation between intervening Si III absorbers and the CGM. Results. Along a total redshift path of Delta z approximate to 24, we identify 69 intervening Si III systems that all show associated absorption from other low and high ions (e.g., H I, Si II, Si IV, C II, C IV). We derive a bias-corrected number density of dN/dz(Si III) = 2.5 +/- 0.4 for absorbers with column densities log N(Si III) > 12.2, which is similar to 3 times the number density of strong Mg II systems at z = 0. This number density matches the expected cross section of a Si III absorbing CGM around the local galaxy population with a mean covering fraction of < f(c)> = 0.69. For the majority (similar to 60 percent) of the absorbers, we identify possible host galaxies within 300 km s(-1) of the absorbers and derive impact parameters rho < 200 kpc, demonstrating that the spatial distributions of Si III absorbers and galaxies are highly correlated. Conclusions. Our study indicates that the majority of Si III-selected absorbers in our sample trace the CGM of nearby galaxies within their virial radii at a typical covering fraction of similar to 70 percent. We estimate that diffuse gas in the CGM around galaxies, as traced by Si III, contains substantially more (more than twice as much) baryonic mass than their neutral interstellar medium.