The charmed strange baryon Ξ0c was searched for in the decay channel Ξ0c → Ξ−π+, and the beauty strange baryon Ξb in the inclusive channel Ξb → Ξ−−¯νX, using the 3.5 million hadronic Z events collected by the DELPHI experiment in the years 1992–1995. The Ξ− was reconstructed through the decay Ξ− → Λπ−, using a constrained fit method for cascade decays. An iterative discriminant analysis was used for the Ξ0c and Ξb selection. The production rates were measured to be fΞ0c ×BR(Ξ0c → Ξ−π+) = (4.7 ± 1.4(stat.) ± 1.1(syst.)) × 10−4 per hadronic Z decay, and BR(b → Ξb)×BR(Ξb → Ξ−−X) = (3.0 ± 1.0(stat.) ± 0.3(syst.)) × 10−4 for each lepton species (electron or muon). The lifetime of the Ξb baryon was measured to be τΞb = 1.45+0.55 −0.43(stat.) ± 0.13(syst.) ps. A combination with the previous DELPHI lifetime measurement gives τΞb = 1.48+0.40 −0.31(stat.) ± 0.12(syst.) ps.