  The production cross-sections of $J/\psi$ mesons in proton-proton collisions
at a centre-of-mass energy of $\sqrt{s}=5$ TeV are measured using a data sample
corresponding to an integrated luminosity of $9.13\pm0.18~\text{pb}^{-1}$,
collected by the LHCb experiment. The cross-sections are measured
differentially as a function of transverse momentum, $p_{\text{T}}$, and
rapidity, $y$, and separately for $J/\psi$ mesons produced promptly and from
beauty hadron decays (nonprompt). With the assumption of unpolarised $J/\psi$
mesons, the production cross-sections integrated over the kinematic range
$0<p_{\text{T}}<20~\text{GeV}/c$ and $2.0<y<4.5$ are
$8.154\pm0.010\pm0.283~\mu\text{b}$ for prompt $J/\psi$ mesons and
$0.820\pm0.003\pm0.034~\mu\text{b}$ for nonprompt $J/\psi$ mesons, where the
first uncertainties are statistical and the second systematic. These
cross-sections are compared with those at $\sqrt{s}=8$ TeV and $13$ TeV, and
are used to update the measurement of the nuclear modification factor in
proton-lead collisions for $J/\psi$ mesons at a centre-of-mass energy per
nucleon pair of $\sqrt{s_{\text{NN}}}=5$ TeV. The results are compared with
theoretical predictions.
