  We investigated the presence of planetary companions around the nearby (7.6
pc) and bright ($V=9$ mag) early-type M dwarf Gl 514, analysing 540 radial
velocities collected over nearly 25 years with the HIRES, HARPS, and CARMENES
spectrographs. The data are affected by time-correlated signals at the level of
2-3 ms$^{-1}$ due to stellar activity, that we filtered out testing three
different models based on Gaussian process regression. As a sanity cross-check,
we repeated the analyses using HARPS radial velocities extracted with three
different algorithms. We used HIRES radial velocities and Hipparcos-Gaia
astrometry to put constraints on the presence of long-period companions, and we
analysed TESS photometric data. We found strong evidence that Gl 514 hosts a
super-Earth on a likely eccentric orbit, residing in the conservative habitable
zone for nearly $34\%$ of its orbital period. The planet Gl 514 b has minimum
mass $m_b\sin i_b=5.2\pm0.9$ $M_{\rm Earth}$, orbital period
$P_b=140.43\pm0.41$ days, and eccentricity $e_b=0.45^{+0.15}_{-0.14}$. No
evidence for transits is found in the TESS light curve. There is no evidence
for a longer period companion in the radial velocities and, based on
astrometry, we can rule out a $\sim0.2$ $M_{\rm Jup}$ planet at a distance of
$\sim3-10$ au, and massive giant planets/brown dwarfs out to several tens of
au. We discuss the possible presence of a second low-mass companion at a
shorter distance from the host than Gl 514 b. Gl 514 b represents an
interesting science case to study the habitability of planets on eccentric
orbits. We advocate for additional spectroscopic follow-up to get more accurate
and precise planetary parameters. Further follow-up is also needed to
investigate sub \ms and shorter period signals.
