Abstract This paper reports the measurement of $$\mathrm{J}/{\psi }$$ J / ψ meson production in proton–proton ( $$\mathrm {p}\mathrm {p}$$ p p ) and proton–lead ( $$\mathrm {p}\mathrm {Pb}$$ p Pb ) collisions at a center-of-mass energy per nucleon pair of $$5.02\,\text {TeV} $$ 5.02 TeV by the CMS experiment at the LHC. The data samples used in the analysis correspond to integrated luminosities of 28 $$\,\text {pb}^{-1}$$ pb - 1 and 35 $$\,\text {nb}^{-1}$$ nb - 1 for $$\mathrm {p}\mathrm {p}$$ p p and $$\mathrm {p}\mathrm {Pb}$$ p Pb collisions, respectively. Prompt and nonprompt $$\mathrm{J}/{\psi }$$ J / ψ mesons, the latter produced in the decay of $${\mathrm {B}}$$ B hadrons, are measured in their dimuon decay channels. Differential cross sections are measured in the transverse momentum range of $$2<p_{\mathrm {T}} <30{\,\text {GeV}/{c}} $$ 2 < p T < 30 GeV / c , and center-of-mass rapidity ranges of $$|y_\mathrm{{CM}} |<2.4$$ | y CM | < 2.4 ( $$\mathrm {p}\mathrm {p}$$ p p ) and $$-2.87<y_\mathrm{{CM}}<1.93$$ - 2.87 < y CM < 1.93 ( $$\mathrm {p}\mathrm {Pb}$$ p Pb ). The nuclear modification factor, $$R_{\mathrm {p}\mathrm {Pb}}$$ R p Pb , is measured as a function of both $$p_{\mathrm {T}}$$ p T and $$y_\mathrm{{CM}}$$ y CM . Small modifications to the $$\mathrm{J}/{\psi }$$ J / ψ cross sections are observed in $$\mathrm {p}\mathrm {Pb}$$ p Pb relative to $$\mathrm {p}\mathrm {p}$$ p p collisions. The ratio of $$\mathrm{J}/{\psi }$$ J / ψ production cross sections in $$\mathrm {p}$$ p -going and Pb-going directions, $$R_\mathrm{{FB}}$$ R FB , studied as functions of $$p_{\mathrm {T}}$$ p T and $$y_\mathrm{{CM}}$$ y CM , shows a significant decrease for increasing transverse energy deposited at large pseudorapidities. These results, which cover a wide kinematic range, provide new insight on the role of cold nuclear matter effects on prompt and nonprompt $$\mathrm{J}/{\psi }$$ J / ψ production.