We report on the extraordinary superionic conductivity in the fulleride polymer Li4C60, a crystalline material with no disorder. 7Li, NMR, and dc frequency dependent conductivity show uncorrelated ionic hopping across small energy barriers (DeltaE_{a} approximately 200 meV) and an ionic conductivity of 10;{-2} S/cm at room temperature, higher than in "standard" ionic conductors. Ab initio calculations of the molecular structure find intrinsic unoccupied interstitial sites that can be filled by Li+ cations in stoichiometric Li4C60 even at low temperatures. The low energy required for the occupation of these sites allows a sizable Li+ diffusion above 130 K. The results suggest novel application of lithium intercalated fullerides as electrodes in Li ions batteries.