Context : Aims : We present a large multi-epoch high resolution spectroscopic investigation for the search of binary candidates in the Galactic Globular Cluster ( GGC ) M4 . The aim of our work is the identification of the binary candidates , and the determination of the binary fraction and of the binary radial distribution . Methods : We have obtained 5973 individual spectra of 2469 stars with the multifiber facility FLAMES+GIRAFFE at VLT . Selected stars go from the red giant branch tip to one magnitude below the TO , and cover the entire cluster . Each star has been observed at least twice , in a temporal interval of about three years . For 484 stars we have three epochs . Radial velocities have been obtained with the classical cross-correlation technique , using a solar type spectrum as template . Results : The average radial velocity of the observed cluster members is 70.29 \pm 0.07 ( \pm 0.3 ) ( \pm 0.1 )  km s ^ { -1 } . The search for variations in radial velocities among the stars with multi-epoch observations yielded 57 binary star candidates . Our radial velocity measurement accuracy allowed us to identify at a 3 \sigma level binaries with radial velocity variations larger than \sim 0.3  km s ^ { -1 } for the target stars with V \leq 15 , and larger than \sim 0.5  km s ^ { -1 } for the targets with V \geq 15 . We identified 4 binary star candidates out of 97 observed targets inside the core radius , and 53 candidates out of 2372 observed stars outside the core radius . Accounting for the incompleteness affecting our survey , the lower limit for the total binary fraction is f = 3.0 \pm 0.3 \% . The lower limit for the binary fraction in the cluster core is f = 5.1 \pm 2.3 \% , while outside the core it decreases to f = 3.0 \pm 0.4 \% . Similarly , we found f = 4.5 \pm 0.4 \% and f = 1.8 \pm 0.6 \% for the binary fraction inside and outside the half mass radius . Conclusions :