In this talk the measurement of the multiplicity dependence of charged pion, kaon, and (anti)proton production in p-Pb collisions at $\sqrt{s_{\rm NN}}\,=5.02$\,TeV at high transverse momenta (<20 GeV/$c$) will be shown, using the ALICE detector at the LHC. The transverse momentum range of the measurement is extended with respect to the already published low-pT results up to 20 GeV/$c$ in seven multiplicity classes, using the capabilities of the High Momentum Particle Identification Detector (HMPID) and the Time Projection Chamber (TPC) in the relativistic rise regime. The presented measurement includes the transverse momentum spectra for $\pi^{\pm}$, K$^{\pm}$ and p($\bar{p}$), the ratios of spectra among different particle species and the nuclear modification factor ($R_{\rm pPb}$) in non-single diffractive p-Pb collisions. The evolution of the spectral shapes and the particle ratios with event multiplicities will be shown and their comparison with pp and Pb-Pb results will be discussed.