We report the detection of a Neptune-mass exoplanet around the M4.0 dwarf GJ 4276 (G 232-070) based on radial velocity (RV) observations obtained with the CARMENES spectrograph. The RV variations of GJ 4276 are best explained by the presence of a planetary companion that has a minimum mass of mb sin i ≈ 16 M⊕ on a Pb = 13.35 day orbit. The analysis of the activity indicators and spectral diagnostics exclude stellar induced RV perturbations and prove the planetary interpretation of the RV signal. We show that a circular single-planet solution can be excluded by means of a likelihood ratio test. Instead, we find that the RV variations can be explained either by an eccentric orbit or interpreted as a pair of planets on circular orbits near a period ratio of 2:1. Although the eccentric single-planet solution is slightly preferred, our statistical analysis indicates that none of these two scenarios can be rejected with high confidence using the RV time series obtained so far. Based on the eccentric interpretation, we find that GJ 4276 b is the most eccentric (eb = 0.37) exoplanet around an M dwarf with such a short orbital period known today. Photometric measurements and Table C.1 are available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (ftp://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/622/A153