This is the third of a series of papers treating light-baryon resonances up to 3 GeV within a relativistically covariant quark model based on the Bethe-Salpeter equation with instantaneous two- and three-body forces. In this last paper we extend our previous work (U. Löring, B.Ch. Metsch, H.R. Petry, this issue, p. 395) on non-strange baryons to a prediction of the complete strange-baryon spectrum and a detailed comparison with experiment. We apply the covariant Salpeter framework, which we developed in the first paper (U. Löring, K. Kretzschmar, B.Ch. Metsch, H.R. Petry, Eur. Phys. J. A 10, 309 (2001)), to the specific quark models introduced in our work published in this issue. Quark confinement is realized by linearly rising three-body string potentials with appropriate Dirac structures; to describe the hyperfine structure of the baryon spectrum we adopt 't Hooft's two-quark residual interaction based on QCD instanton effects. The investigation of instanton-induced effects in the baryon mass spectrum plays a central role in this work. We demonstrate that several prominent features of the excited strange mass spectrum, e.g. the low positions of the strange partners of the Roper resonance or the appearance of approximate “parity doublets” in the Λ spectrum, find a natural, uniform explanation in our relativistic quark model with instanton-induced forces.