  We investigate the stability of four-dimensional dyonic soliton and black
hole solutions of ${\mathfrak {su}}(2)$ Einstein-Yang-Mills theory in anti-de
Sitter space. We prove that, in a neighbourhood of the embedded trivial
(Schwarzschild-)anti-de Sitter solution, there exist non-trivial dyonic soliton
and black hole solutions of the field equations which are stable under linear,
spherically symmetric, perturbations of the metric and non-Abelian gauge field.
