If $U$ is a bounded domain in $\mathbb R^n$ whose boundary is smooth, and $f$ is a smooth function on $U$ whose partial derivatives of all orders have a continuous extensions to $\bar U$. For an arbitrary domain $V \supseteq \bar U$, is there a smooth function $\tilde f$ on $V$ extending $f$?
The case of a half space is a classical result of Seeley (Proc. AMS 15 (1964) 625626), and your case should be just a corollary, using a suitable partition of unity and local change of variables near the boundary. 


You can see the Whitney's Expansion Theorem, whose condition is even weaker, that the domain $A$ of $f$ is merely required as a closed set of $\mathbb{R}^n$, and the extended function is analytic. See Analytic Extensions of Differentiable Functions Defined in Closed Sets. 

