We show that the perturbation of the Su-Schrieffer-Heeger chain by a localized lossy defect leads to higher-order exceptional points (HOEPs). Depending on the location of the defect, third- and fourth-order exceptional points (EP3s & EP4s) appear in the space of Hamiltonian parameters. On the one hand, they arise due to the non-Abelian braiding properties of exceptional lines (ELs) in parameter space. Namely, the HOEPs lie at intersections of mutually non-commuting ELs. On the other hand, we show that such special intersections happen due to the fact that the delocalization of edge states, induced by the non-Hermitian defect, hybridizes them with defect states. These can then coalesce together into an EP3. When the defect lies at the midpoint of the chain, a special symmetry of the full spectrum can lead to an EP4. In this way, our model illustrates the emergence of interesting non-Abelian topological properties in the multiband structure of non-Hermitian perturbations of topological phases.