  We study the decay (at infinity) of extremals of Morrey's inequality in
$\mathbb{R}^n$. These are functions satisfying $$ \displaystyle \sup_{x\neq
y}\frac{|u(x)-u(y)|}{|x-y|^{1-\frac{n}{p}}}= C(p,n)\|\nabla
u\|_{L^p(\mathbb{R}^n)} , $$ where $p>n$ and $C(p,n)$ is the optimal constant
in Morrey's inequality. We prove that if $n \geq 2$ then any extremal has a
power decay of order $\beta$ for any $$
\beta<-\frac13+\frac{2}{3(p-1)}+\sqrt{\left(-\frac13+\frac{2}{3(p-1)}\right)^2+\frac13}.
$$
