Is the operation of inverting a function continuous wrt. this bounded norm?

Question

Let $C:=\big(C^{1,1}(\mathbb{R}^d;\mathbb{R}^d), \|\cdot\|_\infty\big)$ be the space of diffeomorphisms on $\mathbb{R}^d$ with the classical pseudonorm $\|f\|:=\min\!\big(\!\sup_{x\in[0,1]}(|f(x)| + |Df(x)|), 1\big)$.

I was wondering if the 'inversion map' $\iota : C \rightarrow C$ given by $\iota(f):=f^{-1}$ is continuous wrt. $\|\cdot\|$?