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 $12.2pm0.3,mathrm{nb}^{-1}$ for the case where the proton beam is projected into the LHCb detector (corresponding to measuring hadron production at positive rapidity) and $18.6pm0.5,mathrm{nb}^{-1}$ 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.
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