Context : A small number of K-type giants on the red giant branch ( RGB ) is known to be very rich in lithium ( Li ) .
This fact is not accounted for by standard stellar evolution theory .
The exact phase and mechanism of Li enrichment is still a matter of debate .

Aims : Our goal is to probe the abundance of Li along the RGB , from its base to the tip , to confine Li-rich phases that are supposed to occur on the RGB .

Methods : For this end , we obtained medium-resolution spectra with the FLAMES spectrograph at the VLT in GIRAFFE mode for a large sample of 401 low-mass RGB stars located in the Galactic bulge .
The Li abundance was measured in the stars with a detectable Li 670.8 nm line by means of spectral synthesis with COMARCS model atmospheres .
A new 2MASS ( J - K _ { S } ) - T _ { eff } calibration from COMARCS models is presented in the Appendix .

Results : Thirty-one stars with a detectable Li line were identified , three of which are Li-rich according to the usual criterion ( \log \epsilon ( { Li } ) > 1.5 ) .
The stars are distributed all along the RGB , not concentrated in any particular phase of the red giant evolution ( e.g .
the luminosity bump or the red clump ) .
The three Li-rich stars are clearly brighter than the luminosity bump and red clump , and do not show any signs of enhanced mass loss .

Conclusions : We conclude that the Li enrichment mechanism can not be restricted to a clearly defined phase of the RGB evolution of low-mass stars ( M \sim 1 M _ { \sun } ) , contrary to earlier suggestions from disk field stars .