The invention provides golf balls with a core and with one or more layers, preferably cover layers, formed of a saturated rubber/ionomer block copolymer comprising one or more neutral thermoplastic polymer blocks and one or more ionomeric blocks. The saturated rubber block is preferably polyisobutylene, and the ionomer block may be a cationomer block or an anionomer block. Various block cationomers comprising polyisobutylene (PIB) and poly(2-dimethylamino)ethyl methacrylate) (PDMAEMA) are preferred. The golf balls of the present invention have been found to provide improved scuff resistance while providing a greater coefficient of friction for improved spin and feel properties.