  In this manuscript we study rotationally $p$-harmonic maps between spheres.
We prove that for $p\in\mathbb{N}$ given, there exist infinitely many
$p$-harmonic self-maps of $\mathbb{S}^m$ for each $m\in\mathbb{N}$ with $p<m<
2+p+2\sqrt{p}$. In the case of the identity map of $\mathbb{S}^m$ we explicitly
determine the spectrum of the corresponding Jacobi operator and show that for
$p>m$, the identity map of $\mathbb{S}^m$ is stable when interpreted as a
$p$-harmonic self-map of $\mathbb{S}^m$.
