Using a three-dimensional general circulation model, we show that the atmospheric dynamics on a tidally locked Earth-like exoplanet, simulated with the planetary and orbital parameters of Proxima Centauri b, support a longitudinally asymmetric stratospheric wind oscillation (LASO), analogous to Earth's quasi-biennial oscillation (QBO). In our simulations, the LASO has a vertical extent of 35–55 km, a period of 5–6.5 months, and a peak-to-peak wind speed amplitude of −70 to +130 m s−1 with a maximum at an altitude of 41 km. Unlike the QBO, the LASO displays longitudinal asymmetries related to the asymmetric thermal forcing of the planet and to interactions with the resulting stationary Rossby waves. The equatorial gravity wave sources driving the LASO are localized in the deep convection region at the substellar point and in a jet exit region near the western terminator, unlike the QBO, for which these sources are distributed uniformly around the planet. Longitudinally, the western terminator experiences the highest wind speeds and undergoes reversals earlier than other longitudes. The antistellar point only experiences a weak oscillation with a very brief, low-speed westward phase. The QBO on Earth is associated with fluctuations in the abundances of water vapor and trace gases such as ozone, which are also likely to occur on exoplanets if these gases are present. Strong fluctuations in temperature and the abundances of atmospheric species at the terminators will need to be considered when interpreting atmospheric observations of tidally locked exoplanets.