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