  Lithium rich stars in globular clusters are rare. In fact, only 14 have been
found so far, in different evolutionary phases from dwarfs to giants. Different
mechanisms have been proposed to explain this enhancement, but it is still an
open problem. Using spectra collected within the Gaia-ESO Survey, obtained with
the GIRAFFE spectrograph at the ESO Very Large Telescope, we present the
discovery of the first Li-rich star in the cluster NGC 1261, the second star
known in the red giant branch bump phase. The star shows an extreme Li
overabundance of A(Li)_LTE=3.92\pm0.14, corresponding to A(Li)_NLTE=3.40 dex.
We propose that the Li enhancement is caused by fresh Li production through an
extra mixing process (sometimes referred to as {\em cool bottom burning}) or
could be a pre-existing Li overabundance resulting from binary mass transfer,
likely from a red giant branch star, because of the low barium abundance. To
unambiguously explain the Li enhancement in globular cluster stars, however, a
reliable determination of the abundance of key species like Be, 6Li, 12C/13C,
and several s-process elements is required, as well as detailed modeling of
chromospheric activity indicators.
