The decays of the B⁺ meson to the final state D^∗−D<inf>s</inf>⁺π⁺ are studied in proton-proton collision data collected with the LHCb detector at centre-of-mass energies of 7, 8, and 13 TeV, corresponding to a total integrated luminosity of 9 fb⁻¹. The ratio of branching fractions of the B⁺→D^∗−D<inf>s</inf>⁺π⁺ and B⁰→D^∗−D<inf>s</inf>⁺ decays is measured to be 0.173 ± 0.006 ± 0.010, where the first uncertainty is statistical and the second is systematic. Using partially reconstructed D<inf>s</inf>^∗+→D<inf>s</inf>⁺γ and D<inf>s</inf>⁺π⁰ decays, the ratio of branching fractions between the B⁺→D^∗−D<inf>s</inf>^∗+π⁺ and B⁺→D^∗−D<inf>s</inf>⁺π⁺ decays is determined as 1.31 ± 0.07 ± 0.14. An amplitude analysis of the B⁺→D^∗−D<inf>s</inf>⁺π⁺ decay is performed for the first time, revealing dominant contributions from known excited charm resonances decaying to the D^*−π⁺ final state. No significant evidence of exotic contributions in the D<inf>s</inf>⁺π⁺ or D^∗−D<inf>s</inf>⁺ channels is found. The fit fraction of the scalar state T<inf>cs¯0</inf>^∗2900⁺⁺ observed in the B⁺→D⁻D<inf>s</inf>⁺π⁺ decay is determined to be less than 2.3% at a 90% confidence level.