Recently, Borwein and Moors (1998) introduced a new class of tangentially regular sets in IRn (called arc-wise essentially smooth sets). They characterized the sets S of this class in terms of arc-wise essential smoothness of the distance function dS. Very recently, the author (2002) gave an appropriate extension of this class to any Banach space X and he extended the above characterization to any Banach space X with a uniformly Gˆateaux differentiable norm. In this paper we extend the concept of arc-wise essentially smooth sets to set-valued mappings C : [0, T]⇉X (T > 0) and we will use this concept to establish an important application to nonconvex sweeping process.