Let $d\in\mathbb N$, $M$ be a $d$-dimensional properly embedded $C^2$-submanifold of $\mathbb R^d$ with boundary and $\nu_{\partial M}$ denote the outward pointing unit normal field on $\partial M$.
Note that $\nu_{\partial M}$ is $C^1$-differentiable and hence, for any given open neighborhood $O$ of $\partial M$, there is a $\tilde\nu\in C^1(\mathbb R^d,\mathbb R^d)$ with $$\operatorname{supp}\tilde\nu\subseteq O\tag1$$ and $$\left.\tilde\nu\right|_{\partial M}=\nu_{\partial M}\tag2.$$
I've described in this answer how $\tilde\nu$ can be constructed.
Question: Are we able to show that $O$ and $\tilde\nu$ can be chosen such that $$\left\|\tilde\nu(a)\right\|=1\tag3$$ for all $a\in U$ for some open $U\subseteq\mathbb R^d$ with $\partial M\subseteq U\subseteq O$?
Note that $(3)$ is clearly satisfied for all $a\in\partial M$, since $\tilde\nu(a)=\nu_{\partial M}(a)$.
Referring to the notation of the answer in my link, we may be able to chose $\rho_0$ appropriately.
Choose $\bar{\nu},O$ satisfying $(1)$ and $(2)$and define an set $$ U=\left\{x\in\mathbb{R}^d:\|\bar{\nu}(x)\|>\frac{1}{2}\right\} $$ Note that $\partial M\subset U\subseteq O$, and $U$ is open, since it is the preimage of an open set by a continuous function $\|\bar{\nu}\|$. Choose a smooth function function $f:\mathbb{R}\to\mathbb{R}$ such that $f(x)=x^{-1/2}$ for $x\ge 1/4$. Define a vector field $v$ by $$ v(x)=f(\|\bar{\nu}(x)\|^2)\bar{\nu} $$ Then $v$ is as smooth as $\bar{\nu}$ is, $\operatorname{supp}(v)\subseteq O$, $v|_{\partial M}=\nu$, and $\|v|_U\|=1$.