  We establish that the Volterra-type integral operator $J_b$ on the Hardy
spaces $H^p$ of the unit ball $\mathbb{B}_n$ exhibits a rather strong rigid
behavior. More precisely, we show that the compactness, strict singularity and
$\ell^p$-singularity of $J_b$ are equivalent on $H^p$ for any $1 \le p <
\infty$. Moreover, we show that the operator $J_b$ acting on $H^p$ cannot fix
an isomorphic copy of $\ell^2$ when $p \ne 2.$
