Context. It has recently been suggested that all giant stars with masses below 2 M_⊙ suffer an episode of surface lithium enrichment between the tip of the red giant branch (RGB) and the red clump (RC). Aims. We test if the above result can be confirmed in a sample of RC and RGB stars that are members of open clusters. Methods. We discuss Li abundances in six open clusters with ages between 1.5 and 4.9 Gyr (turn-off masses between 1.1 and 1.7 M_⊙). We compare these observations with the predictions of different models that include rotation-induced mixing, thermohaline instability, mixing induced by the first He flash, and energy losses by neutrino magnetic moment. Results. In six clusters, we find close to 35% of RC stars have Li abundances that are similar or higher than those of upper RGB stars. This can be a sign of fresh Li production. Because of the extra-mixing episode connected to the luminosity bump, the expectation has been for RC stars to have systematically lower surface Li abundances. However, we cannot confirm that this possible Li production is ubiquitous. For about 65% of RC giants, we can only determine upper limits in abundances that could be hiding very low Li content. Conclusions. Our results indicate the possibility that Li is being produced in the RC, at levels that would not typically permit the classification of these the stars as Li rich. The determination of their carbon isotopic ratio would help to confirm that the RC giants have suffered extra mixing followed by subsequent Li enrichment. The Li abundances of the RC stars can be qualitatively explained by the models including an additional mixing episode close to the He flash.