In general, there are several candidates for the definition of $C^\infty(K)$:
One is the space $\lbrace f|_K: f\in C^\infty(\mathbb R^n)\rbrace$ of all restrictions (endowed with the quotient topology), another is the intersection  $\bigcap\limits_{k\in\mathbb N_0} \lbrace f|_K: f\in C^k(\mathbb R^n)\rbrace$ (which is equal to the former for $n=1$ due to Merrien but different in general, an elementary example is is Pawlucki's *On the algebra of functions $C^k$-extendable for each $k$ finite*, Proc. Amer. Math. Soc. 133 (2005), no. 2, 481–484), and finally the probably best understood definition is that of Whitney jets, i.e. families $(f^{(\alpha)})_{\alpha \in \mathbb N_0^d}$ of continuous functions which satisfy the correct Taylor approximations on $K$ as if $f^{(\alpha)}=\partial^{\alpha} f$ for some $f\in C^\infty(\mathbb R^n)$. 



If $K$ is the closure of its interior the definitions coincide and you should consult the literature about extension of Whitney jets. The article of Frerick [*Extension operators for spaces of infinite differentiable Whitney jets*, J. Reine Angew. Math. 602 (2007), 123–154] contains a lot of information. As mentioned by Deane Yang Lipschitz boundary is enough for having a continuous linear extension operator (this is due to E.M. Stein). However, a sharp cusp like $K=\lbrace (x,y)\in [0,1]^2: y\le \exp(-1/x)\rbrace$ does not have such an extension.

For general $K$ and the space of all restrictions, the question is wide open, besides the examples of Fefferman and Ricci mentioned by David Roberts there are some results of Vogt [*Restriction spaces of $A^\infty$*, Rev. Mat. Iberoam. 30 (2014), no. 1, 65–78].