Unconventional superconductivity and in particular triplet superconductivity have been front and center of topological materials and quantum technology research. Here we report our observation of triplet superconductivity in nonmagnetic CoSi$_2$/TiSi$_2$ heterostructures on silicon. CoSi$_2$ undergoes a sharp superconducting transition at a critical temperature $T_c approx$ 1.5 K, while TiSi$_2$ is a normal metal. We investigate conductance spectra of both two-terminal CoSi$_2$/TiSi$_2$ tunnel junctions and three-terminal T-shaped CoSi$_2$/TiSi$_2$ superconducting proximity structures. We report an unexpectedly large spin-orbit coupling in CoSi$_2$ heterostructures. Below $T_c$, we observe (1) a narrow zero-bias conductance peak on top of a broad hump, accompanied by two symmetric side dips in the tunnel junctions, (2) a narrow zero-bias conductance peak in T-shaped structures, and (3) hysteresis in the junction magnetoresistance. These three independent and complementary observations are indicative of chiral $p$-wave pairing in CoSi$_2$/TiSi$_2$ heterostructures. This chiral triplet superconductivity and the excellent fabrication compatibility of CoSi$_2$ and TiSi$_2$ with present-day silicon integrated-circuit technology facilitate full scalability for potential use in quantum-computing devices.