Approximation by Lipschitz, Cp smooth functions on weakly compactly generated Banach spaces

It is shown that on weakly compactly generated Banach spaces which admit a Lipschitz, Cp smooth bump function, one can uniformly approximate uniformly continuous, bounded, real-valued functions by Lipschitz, Cp smooth functions. This provides a ‘Lipschitz version’ of the classical approximation results of Godefroy, Troyanski, Whitfield and Zizler.