An amplitude analysis of the B^{+}→D_{s}^{+}D_{s}^{-}K^{+} decay is carried out to study for the first time its intermediate resonant contributions, using proton-proton collision data collected with the LHCb detector at center-of-mass energies of 7, 8, and 13 TeV. A near-threshold peaking structure, referred to as X(3960), is observed in the D_{s}^{+}D_{s}^{-} invariant-mass spectrum with significance greater than 12 standard deviations. The mass, width, and the quantum numbers of the …