A broad peaking structure is observed in the dimuon spectrum of B+ -> K+mu(+)mu(-) 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+mu(+)mu(-) 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(-8)(+9) MeV/c(2) and 65(-16)(+22) MeV/c(2), respectively, where the uncertainties include statistical and systematic contributions. These measurements are compatible with the properties of the psi(4160) meson. First observations of both the decay B+ -> psi(4160)K+ and the subsequent decay psi(4160) -> mu(+)mu(-) are reported. The resonant decay and the interference contribution make up 20% of the yield for dimuon masses above 3770 MeV/c(2). This contribution is larger than theoretical estimates.