  We prove that half spaces are the only stable nonlocal $s$-minimal cones in
$\mathbb{R}^3$, for $s\in(0,1)$ sufficiently close to $1$. This is the first
classification result of stable $s$-minimal cones in dimension higher than two.
Its proof can not rely on a compactness argument perturbing from $s=1$. In
fact, our proof gives a quantifiable value for the required closeness of $s$ to
$1$. We use the geometric formula for the second variation of the fractional
$s$-perimeter, which involves a squared nonlocal second fundamental form, as
well as the recent BV estimates for stable nonlocal minimal sets.
