Engineered spin-orbit coupling (SOC) in cold-atom systems can enable the study of new synthetic materials and complex condensed matter phenomena1,2,3,4,5,6,7,8. However, spontaneous emission in alkali-atom spin-orbit-coupled systems is hindered by heating, limiting the observation of many-body effects1,2,5and motivating research into potential alternatives9,10,11. Here we demonstrate that spin-orbit-coupled fermions can be engineered to occur naturally in a one-dimensional optical lattice clock12. In contrast to previous SOC experiments1,2,3,4,5,6,7,8,9,10,11, here the SOC is both generated and probed using a direct ultra-narrow optical clock transition between two electronic orbital states in 87Sr atoms. We use clock spectroscopy to prepare lattice band populations, internal electronic states and quasi-momenta, and to produce spin-orbit-coupled dynamics. The exceptionally long lifetime of the excited clock state (160 seconds) eliminates decoherence and atom loss from spontaneous emission at all relevant experimental timescales, allowing subsequent momentum- and spin-resolvedin situprobing of the SOC band structure and eigenstates. We use these capabilities to study Bloch oscillations, spin-momentum locking and Van Hove singularities in the transition density of states. Our results lay the groundwork for using fermionic optical lattice clocks to probe new phases of matter.