I have a question about the derivative of a distance function.

Let $D \subset \mathbb{R}^{d}$ be a connected and unbounded open subset with smooth boundary. $B(z,r)$ denotes the open ball of radius $r>0$ centered at $z \in \bar{D}$. We define the following distance function $F$ on $\mathbb{R}^{d}$: \begin{equation*} F:x \mapsto d(x,\partial D \cap B(z,r)). \end{equation*} This function is differentiable in a.e. sense since it is Lipschitz continuous (Rademacher's theorem).

Can we show that the following estimate holds? \begin{equation*} \text{ess inf}_{x \in \mathbb{R}^{d}} |\nabla F(x)|>0 \end{equation*} More weakly, can we show that the following? \begin{equation*} |\nabla F(x)|>0 \text{ a.e.} \end{equation*}

If you know related results, please let me know.