  For $p\in\mathbb{R}$, we show that non-circular closed $p$-elastic curves in
$\mathbb{S}^2$ exist only when $p=2$, in which case they are classical elastic
curves, or when $p\in(0,1)$. In the latter case, we prove that for every pair
of relatively prime natural numbers $n$ and $m$ satisfying $m<2n<\sqrt{2}\,m$,
there exists a closed spherical $p$-elastic curve with non-constant curvature
which winds around a pole $n$ times and closes up in $m$ periods of its
curvature. Further, we show that all closed spherical $p$-elastic curves for
$p\in(0,1)$ are unstable as critical points of the $p$-elastic energy.
