Engineered spin-orbit coupling (SOC) in cold-atom systems can enable the study of new synthetic materials and complex condensed matter phenomena1-8. However, spontaneous emission in alkali-atom spin-orbit-coupled systems is hindered by heating, limiting the observation of many-body effects(1,2,5) and motivating research into potential alternatives(9-11). Here we demonstrate that spin-orbit-coupled fermions can be engineered to occur naturally in a one-dimensional optical lattice clock(12). In contrast to previous SOC experiments(1-11), here the SOC is both generated and probed using a direct ultra-narrow optical clock transition between two electronic orbital states in Sr-87 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-resolved in situ probing 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.