  We report the discovery of HN Lib b, a sub-Neptunian mass planet orbiting the
nearby ($d \approx$ = 6.25 pc) M4.0 V star HN Lib detected by our CARMENES
radial-velocity (RV) survey. We determined a planetary minimum mass of
$M_\text{b}\sin i = $ 5.46 $\pm$ 0.75 $\text{M}_\oplus$ and an orbital period
of $P_\text{b} = $ 36.116 $\pm$ 0.029 d, using $\sim$5 yr of CARMENES data, as
well as archival RVs from HARPS and HIRES spanning more than 13 years. The flux
received by the planet equals half the instellation on Earth, which places it
in the middle of the conservative habitable zone (HZ) of its host star. The RV
data show evidence for another planet candidate with $M_\text{[c]}\sin i = $
9.7 $\pm$ 1.9 $\text{M}_\oplus$ and $P_\text{[c]} = $ 113.46 $\pm$ 0.20 d. The
long-term stability of the signal and the fact that the best model for our data
is a two-planet model with an independent activity component stand as strong
arguments for establishing a planetary origin. However, we cannot rule out
stellar activity due to its proximity to the rotation period of HN Lib, which
we measured using CARMENES activity indicators and photometric data from a
ground-based multi-site campaign as well as archival data. The discovery adds
HN Lib b to the shortlist of super-Earth planets in the habitable zone of M
dwarfs, but HN Lib [c] probably cannot be inhabited because, if confirmed, it
would most likely be an icy giant.
