This talk is about a quantitative stability result for geodesic spheres in rank-one symmetric spaces of non-compact type: real, complex, quaternionic, and octonionic hyperbolic spaces. These manifolds have negatively pinched sectional curvature distributed according to the underlying algebraic structure. In particular, any radial vector field induces a natural comparison map with the real hyperbolic space of the right (real) dimension via exponential coordinates, with an explicit control of the tangential deformations. This is a key point to show that geodesic spheres are isoperimetric among all sets sharing a suitable central symmetry. Furthermore, we show that geodesic spheres are uniformly quantitatively stable under small volume-preserving C1-perturbations. Finally, via a rescaling argument, we give a quantitative proof that, for small volumes, geodesic spheres are the unique isoperimetric regions in rank one symmetric spaces.