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We present Li and Fe abundances for 87 stars in the globular cluster M4, obtained by using high-resolution spectra collected with GIRAFFE at the Very Large Telescope. The targets range from the turn-off up to the red giant branch bump. The Li abundance in the turn-off stars is uniform, with an average value equal to A(Li)= 2.30 ± 0.02 dex ( dex), consistent with the upper envelope of Li content measured in other globular clusters and in the halo field stars, confirming also for M4 the discrepancy with the primordial Li abundance predicted by Wilkinson Microwave Anisotropy Probe+ big bang nucleosynthesis (WMAP+BBNS). The global behaviour of A(Li) as a function of the effective temperature allows us to identify the two main drops in the Li evolution due to the first dredge-up and to the extra-mixing episode after the red giant branch bump. The measured iron content of M4 results to dex ( dex), with no systematic offsets between dwarf and giant stars. The behaviour of the Li and Fe abundances along the entire evolutionary path is incompatible with theoretical models including pure atomic diffusion, pointing out that an additional turbulent mixing below the convective region needs to be taken into account, able to inhibit the atomic diffusion. The measured value of A(Li) and its homogeneity in the turn-off stars allow us to put strong constraints on the shape of the Li profile inside the M4 turn-off stars. The global behaviour of A(Li) with the effective temperature can be reproduced with different pristine Li abundances, depending on the kind of adopted turbulent mixing. One cannot reproduce the global trend that starts from the WMAP+BBNS A(Li) and adopts the turbulent mixing described by Richard, Michaud & Richer with the same efficiency as that used by Korn et al. to explain the Li content in NGC 6397. In fact, such a solution is not able to well reproduce simultaneously the Li abundance observed in turn-off and red giant branch stars. However, the WMAP+BBNS A(Li) can be reproduced assuming a more efficient turbulent mixing able to reach deeper stellar regions where the Li is burned. We conclude that the cosmological Li discrepancy cannot be easily solved with the present, poor understanding of the turbulence in the stellar interiors, and a future effort to well understand the true nature of this non-canonical process is needed. [ABSTRACT FROM AUTHOR]