The multi-strange baryon yields in Pb–Pb collisions have been shown to exhibit an enhancement relative to pp reactions. In this work, and production rates have been measured with the ALICE experiment as a function of transverse momentum, pT, in p–Pb collisions at a centre-of-mass energy of √sNN = 5.02 TeV. The results cover the kinematic ranges 0.6 GeV/c < pT < 7.2 GeV/c and 0.8 GeV/c < pT < 5 GeV/c, for and respectively, in the common rapidity interval −0.5 < yCMS < 0. Multi-strange baryons have been identified by reconstructing their weak decays into charged particles. The pT spectra are analysed as a function of event charged-particle multiplicity, which in p–Pb collisions ranges over one order of magnitude and lies between those observed in pp and Pb–Pb collisions. The measured pT 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 p–Pb collisions. The yield of hyperons relative to charged pions is studied and compared with results from pp and Pb–Pb collisions. A continuous increase in the yield ratios as a function of multiplicity is observed in p–Pb data, the values of which range from those measured in minimum bias pp to the ones in Pb–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.