A broad peaking structure is observed in the dimuon spectrum of $B^+ to K^+ mu^+mu^-$ decays in the kinematic region where the kaon has a low recoil against the dimuon system. The structure is consistent with interference between the $B^+ to K^+ mu^+mu^-$ decay and a resonance and has a statistical significance exceeding six standard deviations. The mean and width of the resonance are measured to be $4191^{+9}_{-8}mathrm{,Mekern -0.1em V/}c^2$ and $65^{+22}_{-16}mathrm{,Mekern -0.1em V/}c^2$, 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 $B^+ to psi(4160) K^+$ and the subsequent decay $psi(4160) to mu^+mu^-$ are reported. The resonant decay and the interference contribution make up 20,% of the yield in the low recoil region, which is larger than theoretical estimates.
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