Let $0 < R \leqslant 1$ be such that $B_R \Subset \Omega \subseteq \mathbb{R}^n$ and let $0< \rho < R$. I am looking for a cut-off function $\eta \in C^{\infty}(\mathbb{R}^n, \mathbb{R}^n)$ such that $0\leqslant \eta \leqslant 1$, $\eta \equiv 1$ on $B_{\rho}$, $\eta \equiv 0$ on $\mathbb{R}^n \smallsetminus B_R$ and $|D\eta| \leqslant \frac{1}{\sqrt{2}}$.
Can you give me an example?
No, you can't always guarantee this for arbitrary $0<\rho<R$. Just think intuitively in 1-dimension: your function is $1$ on $B_{\rho}$, and it has to decrease to $0$ outside $B_R$. So, roughly speaking, the magnitude of the slope of the function between $\rho$ and $R$ is $\left|\frac{0-1}{R-\rho}\right|=\frac{1}{R-\rho}$. So, if $R-\rho$ is very small, then the slope has to exceed $\frac{1}{\sqrt{2}}$. So, the trouble is that if $\rho,R$ are very close together, then your function must very rapidly decrease from $1$ to $0$, hence must have a large slope. I leave it to you to make this argument more precise using the mean-value theorem.
There are definitely variants of theorems regarding bump functions and control over their derivatives. However, such theorems always have something to do with the supports of the functions. As shown above, it is impossible to make uniform-boundedness assertions on the derivatives, without referencing the size of the supports in some way.