The statistical model with exact conservation of baryon number, electric charge, and strangeness – the Canonical Statistical Model (CSM) – is used to analyze the dependence of yields of light nuclei at midrapidity on charged pion multiplicity at the LHC. The CSM calculations are performed assuming baryon-symmetric matter, using the recently developed Thermal-FIST package. The light nuclei-to-proton yield ratios show a monotonic increase with charged pion multiplicity, with a saturation at the corresponding grand-canonical values in the high-multiplicity limit, in good qualitative agreement with the experimental data measured by the ALICE collaboration in pp and Pb–Pb collisions at different centralities and energies. Comparison with experimental data at low multiplicities shows that exact conservation of charges across more than one unit of rapidity and/or a chemical freeze-out temperature which decreases with the charged pion multiplicity improves agreement with the data.