A measurement of light-by-light scattering, $\gamma\gamma \rightarrow \gamma\gamma$, in ultraperipheral PbPb collisions at a centre-of-mass energy per nucleon pair of $5.02$ TeV is reported. The analysis is conducted using a data sample corresponding to an integrated luminosity of $390~\mu \rm{b}^{-1}$ recorded by the CMS experiment at the LHC. Light-by-light scattering processes are selected in events with two photons exclusively produced, each with transverse energy $E_T > 2$ GeV, pseudorapidity $\mid\eta\mid < 2.4$, diphoton invariant mass $m_{\gamma\gamma} > 5$ GeV, diphoton transverse momentum $p_T < 1$ GeV, and diphoton acoplanarity below $0.01$. After all selection criteria are applied, $14$ events are observed, compared to expectations of $11.1 \pm 1.1$ (th) events for the signal and $4.0 \pm 1.2$ (stat) for the background processes. The significance of the light-by-light signal hypothesis against the background-only hypothesis is 4.1 standard deviations. The measured fiducial light-by-light scattering cross section, $\sigma_{fid} (\gamma\gamma \rightarrow \gamma\gamma ) = 120 \pm 46~\rm{(stat)} \pm 29~\rm{(syst)} \pm 4~\rm{(th)}$ nb is consistent with the Standard Model prediction. The present results allow also to place new competitive limits, reported for the first time, on the production of pseudoscalar axion-like particles, produced in the process $\gamma\gamma \rightarrow a \rightarrow \gamma\gamma$, over the mass range $m_a = 5-50$ GeV.