  We report 20 new lithium-rich giants discovered within the Gaia-ESO Survey,
including the first Li-rich giant with evolutionary stage confirmed by CoRoT
data. Atmospheric parameters and abundances were derived in model atmosphere
analyses using medium-resolution GIRAFFE or high-resolution UVES spectra. These
results are part of the fifth internal data release of Gaia-ESO. The Li
abundances were corrected for non-LTE effects. We used Gaia DR2 parallaxes to
estimate distances and luminosities. The giants have A(Li) > 2.2 dex. The
majority of them (14 out of 20 stars) are in the CoRoT fields. Four giants are
located in the field of three open clusters but are not members. Two giants
were observed in fields towards the Galactic bulge but are likely in the inner
disk. One of the bulge field giants is super Li-rich with A(Li) = 4.0 dex. We
identified one giant with infrared excess at 22 microns. Two other giants, with
large vsin i, might be Li-rich because of planet engulfment. Another giant is
found to be barium enhanced and thus could have accreted material from a former
AGB companion. Otherwise, besides the Li enrichment, the evolutionary stages
are the only other connection between these new Li-rich giants. The CoRoT data
confirm that one Li-rich giant is at the core-He burning stage. The other
giants are concentrated in close proximity to the RGB luminosity bump, the
core-He burning stages, or the early-AGB. This is very clear when looking at
the Gaia-based luminosities of the Li-rich giants. This is also seen when the
CoRoT Li-rich giants are compared to a larger sample of 2252 giants observed in
the CoRoT fields by the Gaia-ESO Survey, which are distributed all over the RGB
in the Teff-logg diagram. These observations show that evolutionary stage is a
major factor behind the Li enrichment in giants. Other processes, like planet
accretion, contribute to a smaller scale. [abridged]
