  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.
