A Poincaré–Einstein metric g is called non-degenerate if there are no non-zero infinitesimal Einstein deformations of g, in Bianchi gauge, that lie in $$L^2$$ . We prove that a 4-dimensional Poincaré–Einstein metric is non-degenerate if it satisfies a certain chiral curvature inequality. Write $${{\,\textrm{Rm}\,}}_+$$ for the part of the curvature operator of g which acts on self-dual 2-forms. We prove that if $${{\,\textrm{Rm}\,}}_+$$ is negative definite then g is non-degenerate. This is a chiral generalisation of a result due to Biquard and Lee, that a Poincaré–Einstein metric of negative sectional curvature is non-degenerate.