The multi-strange baryon yields in Pblt="single bond" data-inlimg="/entities/sbnd" src="/sd/grey_pxl.gif" class="glyphImg imgLazyJSB">Pb collisions have been shown to exhibit an enhancement relative to pp reactions. In this work, Ξ and Ω production rates have been measured with the ALICE experiment as a function of transverse momentum, lsi2" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0370269316301745&_mathId=si2.gif&_user=111111111&_pii=S0370269316301745&_rdoc=1&_issn=03702693&md5=16a421cafa0e2a0e956387184f993c7f" title="Click to view the MathML source">pTlass="mathContainer hidden">lass="mathCode">, in plt="single bond" data-inlimg="/entities/sbnd" src="/sd/grey_pxl.gif" class="glyphImg imgLazyJSB">Pb collisions at a centre-of-mass energy of lsi1" class="mathmlsrc">le="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0370269316301745&_mathId=si1.gif&_user=111111111&_pii=S0370269316301745&_rdoc=1&_issn=03702693&md5=22ecafd7b47a44afcb612cc29dfc064b">lass="imgLazyJSB inlineImage" height="15" width="105" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0370269316301745-si1.gif">lass="mathContainer hidden">lass="mathCode">. The results cover the kinematic ranges lsi4" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0370269316301745&_mathId=si4.gif&_user=111111111&_pii=S0370269316301745&_rdoc=1&_issn=03702693&md5=4d11b492fbd8d1bc70d8e60f5649216e" title="Click to view the MathML source">0.6 GeV/c<pT<7.2 GeV/class="mathContainer hidden">lass="mathCode"> and lsi5" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0370269316301745&_mathId=si5.gif&_user=111111111&_pii=S0370269316301745&_rdoc=1&_issn=03702693&md5=b5324ea13fd912edfc17796789fe720f" title="Click to view the MathML source">0.8 GeV/c<pT<5 GeV/class="mathContainer hidden">lass="mathCode">, for Ξ and Ω respectively, in the common rapidity interval lsi6" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0370269316301745&_mathId=si6.gif&_user=111111111&_pii=S0370269316301745&_rdoc=1&_issn=03702693&md5=1ee0a5fb2233cf0f6317de667943088c" title="Click to view the MathML source">−0.5<yCMS<0lass="mathContainer hidden">lass="mathCode">. Multi-strange baryons have been identified by reconstructing their weak decays into charged particles. The lsi2" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0370269316301745&_mathId=si2.gif&_user=111111111&_pii=S0370269316301745&_rdoc=1&_issn=03702693&md5=16a421cafa0e2a0e956387184f993c7f" title="Click to view the MathML source">pTlass="mathContainer hidden">lass="mathCode"> spectra are analysed as a function of event charged-particle multiplicity, which in plt="single bond" data-inlimg="/entities/sbnd" src="/sd/grey_pxl.gif" class="glyphImg imgLazyJSB">Pb collisions ranges over one order of magnitude and lies between those observed in pp and Pblt="single bond" data-inlimg="/entities/sbnd" src="/sd/grey_pxl.gif" class="glyphImg imgLazyJSB">Pb collisions. The measured lsi2" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0370269316301745&_mathId=si2.gif&_user=111111111&_pii=S0370269316301745&_rdoc=1&_issn=03702693&md5=16a421cafa0e2a0e956387184f993c7f" title="Click to view the MathML source">pTlass="mathContainer hidden">lass="mathCode"> distributions are compared to the expectations from a Blast-Wave model. The parameters which describe the production of lighter hadron species also describe the hyperon spectra in high multiplicity plt="single bond" data-inlimg="/entities/sbnd" src="/sd/grey_pxl.gif" class="glyphImg imgLazyJSB">Pb collisions. The yield of hyperons relative to charged pions is studied and compared with results from pp and Pblt="single bond" data-inlimg="/entities/sbnd" src="/sd/grey_pxl.gif" class="glyphImg imgLazyJSB">Pb collisions. A continuous increase in the yield ratios as a function of multiplicity is observed in plt="single bond" data-inlimg="/entities/sbnd" src="/sd/grey_pxl.gif" class="glyphImg imgLazyJSB">Pb data, the values of which range from those measured in minimum bias pp to the ones in Pblt="single bond" data-inlimg="/entities/sbnd" src="/sd/grey_pxl.gif" class="glyphImg imgLazyJSB">Pb collisions. A statistical model qualitatively describes this multiplicity dependence using a canonical suppression mechanism, in which the small volume causes a relative reduction of hadron production dependent on the strangeness content of the hyperon.