Solitons are ubiquitous in many areas of physics and the natural sciences. Quantum dynamics of solitons, however, has proven experimentally challenging to measure. We show that discrete solitons (“kinks”) can be realized in existing ion traps. These solitons carry a localized, high-frequency and gap-separated internal mode. Accounting for all nonlinear interactions at the Doppler temperature, we show that coherence persists for many periods of oscillation of this mode. We suggest an experiment which could allow a first direct measurement of coherent dynamics of discrete solitons, and explore ideas for the utilization of solitons as carriers of quantum information.