We developed a far-infrared Fabry-Perot filter constructed from a single silicon substrate. The limiting resolving power caused by beam divergence of a silicon-gap Fabry-Perot filter is approximately 10 times higher than that of a vacuum-gap Fabry-Perot filter because of the large index of refraction of silicon. The filter thus permits compact, high-throughput optical systems. Metal mesh patterns microlithographed on each face provide enhanced, wavelength-dependent face reflectivity. We tested the performance of filters with metal mesh patterns consisting of inductive crosses and capacitive squares. A Fabry-Perot filter developed for a rocketborne astrophysics experiment with a capacitive square metal mesh pattern achieves a resolving power of λ/Δλ(FWHM) =160 at λ= 158 µm, with a peak transmittance of 37% over an active aperture of 6.9 mm for an ƒ/3.8 optical beam at 15° incidence. The absorptivity of a 240-µm thick silicon substrate patterned with capacitive metal mesh is A ≲ 1% per pass, including loss in both the silicon and the metal mesh.