A broad peaking structure is observed in the dimuon spectrum of B+→K+μ+μ− decays in the kinematic region where the kaon has a low recoil against the dimuon system. The structure is consistent with interference between the B+→K+μ+μ− decay and a resonance and has a statistical significance exceeding six standard deviations. The mean and width of the resonance are measured to be 4191+9−8 MeV/c2 and 65+22−16 MeV/c2, respectively, where the uncertainties include statistical and systematic contributions. These measurements are compatible with the properties of the ψ(4160) meson. First observations of both the decay B+→ψ(4160)K+ and the subsequent decay ψ(4160)→μ+μ− are reported. The resonant decay and the interference contribution make up 20% of the yield for dimuon masses above 3770 MeV/c2. This contribution is larger than theoretical estimates.