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The angle between the spin of a star and the orbital planes of its planets traces the history of the planetary system. Exoplanets orbiting close to cool stars are expected to be on circular, aligned orbits because of strong tidal interactions with the stellar convective envelope. Spin–orbit alignment can be measured when the planet transits its star, but such ground-based spectroscopic measurements are challenging for cool, slowly rotating stars. Here we report the three-dimensional characterization of the trajectory of an exoplanet around an M dwarf star, derived by mapping the spectrum of the stellar photosphere along the chord transited by the planet. We find that the eccentric orbit of the Neptune-mass exoplanet GJ 436b is nearly perpendicular to the stellar equator. Both eccentricity and misalignment, surprising around a cool star, can result from dynamical interactions (via Kozai migration) with a yet-undetected outer companion. This inward migration of GJ 436b could have triggered the atmospheric escape that now sustains its giant exosphere.