  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.
