Context :

Aims : We report the discovery of two Li-rich giants , with A ( Li ) \sim 1.50 , in an analysis of a sample of 40 giants of the open cluster Trumpler 20 ( with turnoff mass \sim 1.8 M _ { \sun } ) .
The cluster was observed in the context of the Gaia-ESO Survey .

Methods : The atmospheric parameters and Li abundances were derived using high-resolution UVES spectra .
The Li abundances were corrected for nonlocal thermodynamical equilibrium ( non-LTE ) effects .

Results : Only upper limits of the Li abundance could be determined for the majority of the sample .
Two giants with detected Li turned out to be Li rich : star MG 340 has A ( Li ) _ { non - LTE } = 1.54 \pm 0.21 dex and star MG 591 has A ( Li ) _ { non - LTE } = 1.60 \pm 0.21 dex .
Star MG 340 is on average \sim 0.30 dex more rich in Li than stars of similar temperature , while for star MG 591 this difference is on average \sim 0.80 dex .
Carbon and nitrogen abundances indicate that all stars in the sample have completed the first dredge-up .

Conclusions : The Li abundances in this unique sample of 40 giants in one open cluster clearly show that extra mixing is the norm in this mass range .
Giants with Li abundances in agreement with the predictions of standard models are the exception .
To explain the two Li-rich giants , we suggest that all events of extra mixing have been inhibited .
This includes rotation-induced mixing during the main sequence and the extra mixing at the red giant branch luminosity bump .
Such inhibition has been suggested in the literature to occur because of fossil magnetic fields in red giants that are descendants of main-sequence Ap-type stars .