Using a proton-proton collision data sample corresponding to an integrated luminosity of 3 fb−1 collected by LHCb at center-of-mass energies of 7 and 8 TeV, about 3800 Ξ0b→Ξ+cπ−, Ξ+c→pK−π+ signal decays are reconstructed. From this sample, the first measurement of the Ξ0b baryon lifetime is made, relative to that of the Λ0b baryon. The mass differences M(Ξ0b)−M(Λ0b) and M(Ξ+c)−M(Λ+c) are also measured with precision more than four times better than the current world averages. The resulting values are τΞ0bτΛ0b=1.006±0.018±0.010, M(Ξ0b)−M(Λ0b)=172.44±0.39±0.17MeV/c2, M(Ξ+c)−M(Λ+c)=181.51±0.14±0.10MeV/c2, where the first uncertainty is statistical and the second is systematic. The relative rate of Ξ0b to Λ0b baryon production is measured to be fΞ0bfΛ0b(Ξ0b→Ξ+cπ−)(Λ0b→Λ+cπ−)(Ξ+c→pK−π+)(Λ+c→pK−π+)=(1.88±0.04±0.03)×10−2, where the first factor is the ratio of fragmentation fractions, b→Ξ0b relative to b→Λ0b. Relative production rates as functions of transverse momentum and pseudorapidity are also presented.