Context : Aims : We report on radial velocity time series for two M0.0 V stars , GJ 338 B and GJ 338 A , using the CARMENES spectrograph , complemented by ground-telescope photometry from Las Cumbres and Sierra Nevada observatories . We aim to explore the presence of small planets in tight orbits using the spectroscopic radial velocity technique . Methods : We obtained 159 and 70 radial velocity measurements of GJ 338 B and A , respectively , with the CARMENES visible channel between 2016 January and 2018 October . We also compiled additional relative radial velocity measurements from the literature and a collection of astrometric data that cover 200 a of observations to solve for the binary orbit . Results : We found dynamical masses of 0.64 \pm 0.07 M _ { \odot } for GJ 338 B and 0.69 \pm 0.07 M _ { \odot } for GJ 338 A . The CARMENES radial velocity periodograms show significant peaks at 16.61 \pm 0.04 d ( GJ 338 B ) and 16.3 ^ { +3.5 } _ { -1.3 } d ( GJ 338 A ) , which have counterparts at the same frequencies in CARMENES activity indicators and photometric light curves . We attribute these to stellar rotation . GJ 338 B shows two additional , significant signals at 8.27 \pm 0.01 and 24.45 \pm 0.02 d , with no obvious counterparts in the stellar activity indices . The former is likely the first harmonic of the star ’ s rotation , while we ascribe the latter to the existence of a super-Earth planet with a minimum mass of 10.27 ^ { +1.47 } _ { -1.38 } M _ { \oplus } orbiting GJ 338 B . We have not detected signals of likely planetary origin around GJ 338 A . Conclusions : GJ 338 Bb lies inside the inner boundary of the habitable zone around its parent star . It is one of the least massive planets ever found around any member of stellar binaries . The masses , spectral types , brightnesses , and even the rotational periods are very similar for both stars , which are likely coeval and formed from the same molecular cloud , yet they differ in the architecture of their planetary systems .