When electrons are confined in two-dimensional materials, quantum-mechanical transport phenomena and high mobility can be observed. Few demonstrations of these behaviours in surface spin-orbit bands exist. Here, we report the observation of quantized conductance in the surface bands of 50-nm Bi nanowires. With increasing magnetic fields oriented along the wire axis, the wires exhibit a stepwise increase in conductance and oscillatory thermopower, possibly due to an increased number of high-mobility spiral surface modes based on spin-split bands. Surface high mobility is unexpected since bismuth is not a topological insulator and the surface is not suspended but in contact with the bulk. The oscillations enable us to probe the surface structure. We observe that mobility increases dramatically with magnetic fields because, owing to Lorentz forces, spiral modes orbit decreases in diameter pulling the charge carriers away from the surface. Our mobility estimates at high magnetic fields are comparable, within order of magnitude, to the mobility values reported for suspended graphene. Our findings represent a key step in understanding surface spin-orbit band electronic transport.