Multiplicity dependence of pion, kaon, proton and lambda production in p–Pb collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msqrt><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">NN</mml:mi></mml:mrow></mml:msub></mml:msqrt><mml:mo>=</mml:mo><mml:mn>5.02</mml:mn><mml:mtext> TeV</mml:mtext></mml:math>
In this Letter, comprehensive results on ${\rm\pi}^\pm$, K$^\pm$, K$^0_S$, p, $\rm\bar{p}$, $\rm \Lambda$ and $\rm \bar{\Lambda}$ production at mid-rapidity ($0 < y_{\rm cms} < 0.5$) in p-Pb collisions at $\sqrt{s_{\rm NN}} = 5.02$ TeV, measured by the ALICE detector at the LHC, are reported. The transverse momentum distributions exhibit a hardening as a function of event multiplicity, which is stronger for heavier particles. This behavior is similar to what has been observed in pp and Pb-Pb collisions at the LHC. The measured $p_{\rm T}$ distributions are compared to results at lower energy and with predictions based on QCD-inspired and hydrodynamic models.