The measurement of the deuteron and anti-deuteron production in the rapidity range $-1<y<0$ as a function of transverse momentum and event multiplicity in p–Pb collisions at $\sqrt{s_{\mathrm{NN}}}$ = 5.02 TeV is presented. (Anti-)deuterons are identified via their specific energy loss $dE/dx$ and via their time-of-flight. Their production in p–Pb collisions is compared to pp and Pb–Pb collisions and is discussed within the context of thermal and coalescence models. The ratio of integrated yields of deuterons to protons (d/p) shows a significant increase as a function of the charged-particle multiplicity of the event starting from values similar to those observed in pp collisions at low multiplicities and approaching those observed in Pb–Pb collisions at high multiplicities. The mean transverse particle momenta are extracted from the deuteron spectra and the values are similar to those obtained for p and Λ particles. Thus, deuteron spectra do not follow mass ordering. This behaviour is in contrast to the trend observed for non-composite particles in p–Pb collisions. In addition, the production of the rare ${}^3\mathrm{He}$ and ${}^3\overline{\mathrm{He}}$ nuclei has been studied. The spectrum corresponding to all non-single diffractive p-Pb collisions is obtained in the rapidity window $-1<y<0$ and the $p_T$-integrated yield dN/dy is extracted. It is found that the yields of protons, deuterons, and ${}^3\mathrm{He}$, normalised by the spin degeneracy factor, follow an exponential decrease with mass number.