We construct and studySU (3) chiral bags (called chiral hyperbags) in the scheme of collective-coordinate quantization with chiral symmetry breaking treated as perturbation. We show how the Wess-Zumino constraint arises from the quark-bag sector, complementing the soliton sector, in a manner analogous to what happens in (1+1) dimensional chiral bags. Due to the Wess-Zumino term, all the quantum numbers — baryon charge, isopin, angular momentum, hypercharge etc. are fractionized in a prescribed manner. One notable aspect of the fractionization is that for all ranges of bag radius, there is alwaysmore angular momentum lodged in the soliton sector than in the quark sector. It is shown thatwithin the scheme we have adopted, the symmetry breaking termobstructs the Cheshire Cat principle and that consequently when strange quarks are present, the baryons (i.e. hyperons) favor a bigger bag (say R ⪆ 1 fm) than non-strange baryons; this confirms a phenomenological argument put forward some time ago by Brown, Klimpt, Rho and Weise (at least in the collective-coordinate scheme). Our approach allows us to calculate the strangeness content of the proton — a highly topical issue — and we find that while a perturbative treatment of the symmetry breaking term can be made to work (for a big bag) for hyperon spectroscopy, the strangeness content of the proton is insensitive to the bag radius; for relevant ranges of bag radius, the ¯ss admixture stays significant, say, ⪆19%. This result is in stark contrast to the Callan-Klebanov Skyrmion — a remarkably successful model for hyperons — which predicts only about 3%. A subtle role of the Wess-Zumino term is suggested.