I am aware of the inverse function theorem for Lipschitz maps, which uses the notion of generalised derivative $\delta_{x_0} f$ of a Lipschitz map $f$, due to F.H.Clarke in On the inverse function theorem:

**Theorem.** Let $f : U \subset \mathbb{R}^n \to \mathbb{R}^n$ be Lipschitz, and suppose that every matrix $A ∈ \delta_{x_0}f$
is invertible. Then there are neighbourhoods $U$ of $x_0$ and $V$ of $f(x_0)$ in $\mathbb{R}^n$ and a Lipschitz map $g : V → U$ such that $f \circ g = \mathbb{1}_V$ and $g \circ f = \mathbb{1}_U$.

I am looking for a reference to an inverse function theorem in the similar setting which provides a bound for the inner radius of the neighbourhoods $U$ and $V$.

The closest I have been able to find is due to Ralph Howard's note The inverse function theorem for Lipschitz maps. In the case of $B_r$ ball of radius $r$ centered in zero his result reads:

**Theorem.** Let $f : \overline{B_r} \subset \mathbb{R}^n \to \mathbb{R}^n$, $f(O)=O$, $L:\mathbb{R}^n \to \mathbb{R}^n$ invertible linear map and $\rho<1$ such that
$$\forall x_1,x_2\in \overline{B_r},\quad \lVert L^{-1}(f(x_2)-f(x_1)) - (x_2-x_1) \rVert \leq \rho \lVert x_2 - x_1 \rVert\,.$$
Then $f$ is injective on $\overline{B_r}$ with
$$\forall x_1,x_2\in \overline{B_r},\quad \frac{1-\rho}{\lVert L^{-1}\rVert} \lVert x_2-x_1\rVert \leq \lVert f(x_2)-f(x_1) \rVert \leq \lVert L\rVert (1+\rho) \lVert x_2-x_1 \rVert \,.$$
Furthermore, $f$ restricted to open ball $B_r$ is lipeomorphism onto $V=f[B_r]$, where $V$ contains the open ball $B_{r_1}$ of radius $r_1=\rho\frac{1-\rho}{\lVert L^{-1}\rVert}$.

In the case of differentiable $f$, results related to what I am looking for are usually based on Newton's iterates. For example, see Theorem 4.1 of Xinghua, Convergence of Newton's method and inverse function theorem in Banach space.