Let $P\colon U\subset \mathbb{R}^K\rightarrow \mathbb{R}^K$ be $C^\infty$, $\Omega \subset \mathbb{R}^d$ open and bounded, $m \in \mathbb{N}$, $p\in [1,\infty]$ such that $m>\frac{d}{p}$, $f \in W^{m,p}(\Omega,U)$ with $P(f)=f$, $\epsilon>0$ and $l\leq m$. I want to prove that there exist $h>0$ such that
$$|P(g_1)-P(g_2)|_{W^{l,p}}\leq (1+\epsilon)|g_1-g_2|_{W^{l,p}}$$
for all functions $g_1,g_2 \in W^{m,p}(\Omega,U)$ with $|g_1-f|_{W^{m,p}}\leq h$ and $|g_2-f|_{W^{m,p}}\leq h$.
In my application $P=P_M\colon U \rightarrow M$ is the shortest point projection onto a compact submanifold $M \subset \mathbb{R}^K$. I could find similar results, but not the one I need, in section 5.2.2 of

*Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equation* 
by Thomas Runst and Winfried Sickel

My goal is to extend error estimates for an operator $Q$ to error estimates of the operator $P_M \circ Q$ for functions with values in $M$.