Feynman integrals of $p$-adic argument in momentum space III. Renormalization

The ultraviolet and infrared sequences of the poles for the massless Feynman integrals of $p$-adic argument with the arbitrary power of propagators are found. Within the framework of MS-scheme for the analytic $R^*$-operation it is proved that the renormalized amplitudes are finite. Simple formulas for the calculation of counterterms (vertex parts) in coordinate and momentum spaces are given.