The open heavy-flavour hadron measurements in proton--proton and proton--lead collisions give insight into the charm production and hadronization mechanisms. In this contribution, the latest measurements of the production cross section of prompt $\Lambda^+_{\rm c}$ and its charge conjugate performed with the ALICE detector at midrapidity in pp, and the new measurement of $\Lambda_\mathrm{c}^+ \to \mathrm{p} \mathrm{K}_\mathrm{S}^0$ performed down to $p_{\rm T}=0$ in p--Pb collisions at $\sqrt{s_{\rm NN}}$ = 5.02 TeV are presented. We also present the first ALICE measurement of the baryon-to-meson ratio $\Lambda^+_{\rm c}/{\rm D^0}$ and the $\Lambda^+_{\rm c}$ nuclear modification factor $R_{\rm pPb}$ down to $p_{\rm T}$ = 0 in p--Pb collisions. The $\Lambda^+_{\rm c}/{\rm D^0}$ ratio at midrapidity at the LHC is significantly higher than the one in ${\rm e^+e^-}$ collisions, suggesting that the fragmentation of charm is not universal across different collision systems. The results are compared with theoretical calculations.