Context. Teegarden’s Star is the brightest and one of the nearest ultra-cool dwarfs in the solar neighbourhood. For its late spectral type (M7.0 V), the star shows relatively little activity and is a prime target for near-infrared radial velocity surveys such as CARMENES. Aims. As part of the CARMENES search for exoplanets around M dwarfs, we obtained more than 200 radial-velocity measurements of Teegarden’s Star and analysed them for planetary signals. Methods. We find periodic variability in the radial velocities of Teegarden’s Star. We also studied photometric measurements to rule out stellar brightness variations mimicking planetary signals. Results. We find evidence for two planet candidates, each with 1.1 M_⊕ minimum mass, orbiting at periods of 4.91 and 11.4 d, respectively. No evidence for planetary transits could be found in archival and follow-up photometry. Small photometric variability is suggestive of slow rotation and old age. Conclusions. The two planets are among the lowest-mass planets discovered so far, and they are the first Earth-mass planets around an ultra-cool dwarf for which the masses have been determined using radial velocities.