The rigorous QED evaluation of the one- and two-photon exchange corrections to the ground-state hyperfine splitting in Li-like ions is presented for the wide range of nuclear charge number Z=7-82. The calculations are carried out in the framework of the extended Furry picture, i.e., with the inclusion of the effective local screening potential in the zeroth-order approximation. The interelectronic-interaction contributions of the third and higher orders are taken into account in the framework of the Breit approximation employing the recursive perturbation theory. In comparison to the previous theoretical calculations, the accuracy of the interelectronic-interaction contributions to the ground-state hyperfine splitting in Li-like ions is substantially improved.