  We study the sharp constant in the Morrey inequality for fractional
Sobolev-Slobodecki\u{\i} spaces on the whole $\mathbb{R}^N$. By generalizing a
recent work by Hynd and Seuffert, we prove existence of extremals, together
with some regularity estimates. We also analyze the sharp asymptotic behaviour
of this constant as we reach the borderline case $s\,p=N$, where the inequality
fails. This can be done by means of a new elementary proof of the Morrey
inequality, which combines: a local fractional Poincar\'e inequality for
punctured balls, the definition of capacity of a point and Hardy's inequality
for the punctured space. Finally, we compute the limit of the sharp Morrey
constant for $s\nearrow 1$, as well as its limit for $p\nearrow \infty$. We
obtain convergence of extremals, as well.
