Differentiable operators of nearly best approximation

Let $X$ be a normed linear space, let $Y\subset X$ be a finite-dimensional subspace, and let $\varepsilon>0$. We define a multiplicative $\varepsilon$-selection $M\colon X\to Y$ to be a map such that
$$ \forall\,x\in X \qquad \|Mx-x\|\leqslant \inf\{\|x-y\|\colon y\in Y\}(1+\varepsilon). $$

We prove that there is an $\varepsilon$-selection $M$ whose smoothness coincides with that of the norm in $X$. We show that, generally speaking, it is impossible to find an $\varepsilon$-selection of greater smoothness in $L^p[0,1]$.