  The production of $B^+$, $B^0$ and $\Lambda_b^0$ hadrons is studied in
proton-lead collisions at a centre-of-mass energy per nucleon pair of
$\sqrt{s_\mathrm{NN}}=8.16\,{\rm TeV}$ recorded with the LHCb detector at the
LHC. The measurement uses a dataset corresponding to an integrated luminosity
of $12.2\pm0.3\,\mathrm{nb}^{-1}$ for the case where the proton beam is
projected into the LHCb detector (corresponding to measuring hadron production
at positive rapidity) and $18.6\pm0.5\,\mathrm{nb}^{-1}$ for the lead beam
projected into the LHCb detector (corresponding to measuring hadron production
at negative rapidity). Nuclear effects are probed through double-differential
cross-sections, forward-to-backward cross-section ratios and nuclear
modification factors of the beauty hadrons. The double-differential
cross-sections are measured as a function of the beauty-hadron transverse
momentum and rapidity in the nucleon-nucleon centre-of-mass frame.
Forward-to-backward cross-section ratios and nuclear modification factors
indicate a significant nuclear suppression at positive rapidity. The ratio of
$\Lambda_b^0$ over $B^0$ production cross-sections is reported and is
consistent with the corresponding measurement in $pp$~collisions.
