Context. The abundance inhomogeneities of light elements observed in Globular Clusters (GCs), and notably the ubiquitous Na-O anti-correlation, are generally interpreted as evidence that GCs comprise several generations of stars. There is an on-going debate as to the nature of the stars which produce the inhomogeneous elements, and investigating the behavior of several elements is a way to shed new light on this problem. Aims. We aim at investigating the Li and Na content of the GC M 4, that is known to have a well defined Na-O anti-correlation. Methods. We obtained moderate resolution (R=17 000-18 700) spectra for 91 main sequence (MS)/sub-giant branch stars of M 4 with the Giraffe spectrograph at the FLAMES/VLT ESO facility. Using model atmospheres analysis we measured lithium and sodium abundances. Results. We detect a weak Li-Na anti-correlation among un-evolved MS stars. One star in the sample, # 37934, shows the remarkably high lithium abundance A(Li)=2.87, compatible with current estimates of the primordial lithium abundance. Conclusions. The shallow slope found for the Li-Na anti-correlation suggests that lithium is produced in parallel to sodium. This evidence, coupled with its sodium-rich nature, suggests that the high lithium abundance of star # 37934 may originate by pollution from a previous generations of stars. The recent detection of a Li-rich dwarf of pollution origin in the globular cluster NGC 6397 may also point in this direction. Still, no clear cut evidence is available against a possible preservation of the primordial lithium abundance for star # 37934.
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