Let X be an infinite-dimensional Banach space, and let $B_{X}$ and $S_{X}$
be its closed unit ball and unit sphere, respectively. A continuous mapping $%
R:B_{X}\rightarrow S_{X}$ is said to be a retraction provided that $x=Rx$
for all $x\epsilon S_{X}$. It is well known that when X is
finite-dimensional there is no retraction from $B_{X}$ onto $S_{X}.$ We
prove that in some Banach spaces of continuous functions for every $%
\varepsilon >0$ there exists a retraction of the closed unit ball onto the
unit sphere being a $(1+\varepsilon )$-set contraction.