The production of $\Lambda^+_c$ baryons produced directly at the interacting point is studied in proton-lead collisions collected with the LHCb detector at the LHC. The data sample corresponds to an integrated luminosity of $1.58\mathrm{nb}^{-1}$ recorded at a nucleon-nucleon centre-of-mass energy of $\sqrt{s_{NN}}=5.02$ TeV. Measurements of the differential cross-section and the forward-backward production ratio are reported for $\Lambda^+_c$ baryons with transverse momenta in the range $2<p_{T}<10$GeV/$c$ and rapidities in the ranges $1.5<y^*<4.0$ and $-4.5<y^*<-2.5$ in the nucleon-nucleon centre-of-mass system. The ratio of cross-sections of $\Lambda^+_c$ baryons and $D^0$ mesons is also reported. The results are compared with next-to-leading order calculations that use nuclear parton distribution functions.