A measurement of CP-violating observables in B^± → D^*K^± and B^± → D^*π^± decays is made where the photon or neutral pion from the D^*→ Dγ or D^*→ Dπ⁰ decay is not reconstructed. The D meson is reconstructed in the self-conjugate decay modes, D → K<inf>S</inf>⁰π⁺π⁻ or D → K<inf>S</inf>⁰K⁺K⁻. The distribution of signal yields in the D decay phase space is analysed in a model-independent way. The measurement uses a data sample collected in proton-proton collisions at centre-of-mass energies of 7, 8, and 13 TeV, corresponding to a total integrated luminosity of approximately 9 fb⁻¹. The B^± → D^*K^± and B^± → D^*π^±CP-violating observables are interpreted in terms of hadronic parameters and the CKM angle γ, resulting in a measurement of γ = (92<inf>−17</inf>⁺²¹)^°. The total uncertainty includes the statistical and systematic uncertainties, and the uncertainty due to external strong-phase inputs.