The band structures of the doubly-odd nucleus 116I have been studied by using in-beam γ-ray spectroscopy with the 103Rh( 16O, 3n) 116I reaction at an energy of E lab=80 MeV. The beam was provided by the 12UD tandem accelerator at the University of Tsukuba. Absolute excitation energies for the observed collective bands except for a band based on the π(h 11/2) ν(g 7/2) configuration have been definitely determined. The positive-parity rotational band based on the π(h 11/2) ν(h 11/2) configuration is built on the 8 + level at 874.3 keV and exhibits a spin inversion below I=14ℏ. A new vibration-like band built on the 8 − level at 468.5 keV has been identified and can be interpreted as resulting from the π(g 7/2) ν(h 11/2) configuration coupled to the Te core vibration. A negative-parity band built on the 14 − level at 2547.6 keV shows a vibration-like structure and terminates its collectivity at the 24 − level at 6215.0 keV. The total Routhian surface calculations suggest a noncollective oblate state with β 2=0.182 and γ=60° based on a fully aligned six-quasiparticle, π[h 11/2(g 7/2) 2] ν[g 7/2(h 11/2) 2], configuration for this 24 − level. The excitation energy of the 7 − isomer with T 1/2=3.27 μs was found to be at 430.4 keV.