Smoothing of Hilbert-valued uniformly continuous maps

We approximate (in the uniform norm) Hilbert-valued uniformly continuous maps defined on $l_p$, $p\geqslant2$, by maps with bounded first derivatives and maximal local smoothness, which coincides with the smoothness of the space. The result obtained is definitive as far as the smoothness of smoothing maps is concerned since there is a 1-Lipschitzian map from $l_p$, $p\geqslant2$, to $l_2$ that cannot be approximated in the uniform metric by a map whose first derivative is uniformly continuous.