  We prove that in any rank one symmetric space of non-compact type
$M\in\{\mathbb{R} H^n,\mathbb{C} H^m,\mathbb{H} H^m,\mathbb{O} H^2\}$, geodesic
spheres are uniformly quantitatively stable with respect to small $C^1$-volume
preserving perturbations. We quantify the gain of perimeter in terms of the
$W^{1,2}$-norm of the perturbation, taking advantage of the explicit spectral
gap of the Laplacian on geodesic spheres in $M$. As a consequence, we give a
quantitative proof that for small volumes, geodesic spheres are isoperimetric
regions among all sets of finite perimeter.
