In infinite dimensional Banach spaces the unit sphere is a lipschitzian retract of the unit ball. We use the space of continuous functions vanishing at a point to provide an example of such retraction having relatively small Lipschitz constant.
"Retracting ball onto sphere in $BC_0(\mathbb R)$." Topol. Methods Nonlinear Anal. 33 (2) 307 - 313, 2009.