The production cross sections of the ${B}^{+}$, ${B}^{0}$, and ${B}_{s}^{0}$ mesons, and of their charge conjugates, are measured via exclusive hadronic decays in $p+\mathrm{Pb}$ collisions at the center-of-mass energy $\sqrt{{s}_{NN}}=5.02\text{ }\text{ }\mathrm{TeV}$ with the CMS detector at the CERN LHC. The data set used for this analysis corresponds to an integrated luminosity of $34.6\text{ }\text{ }{\mathrm{nb}}^{\ensuremath{-}1}$. The production cross sections are measured in the transverse momentum range between 10 and $60\text{ }\text{ }\mathrm{GeV}/c$. No significant modification is observed compared to proton-proton perturbative QCD calculations scaled by the number of incoherent nucleon-nucleon collisions. These results provide a baseline for the study of in-medium $b$ quark energy loss in $\mathrm{Pb}+\mathrm{Pb}$ collisions.