High-spin states in neutron-rich indium isotope $^{116}$In have been observed for the first time by using incomplete fusion of $^{7}$Li on a $^{116}$Cd target at a beam energy of 42 MeV. A negative-parity band with $\pi g_{9/2}^{-1}\otimes \nu h_{11/2}$ configuration and a pair of positive-parity doublet bands with $\pi g_{9/2}^{-1}\otimes \nu (g_{7/2}/d_{5/2})h_{11/2}^2$ configurations have been identified. The positive-parity doublet bands are tentatively interpreted as chiral doublet bands, which is supported by the adiabatic and configuration-fixed constrained triaxial relativistic-mean-field and multiparticle plus rotor model calculations. This observation reports a possible new region of chirality in neutron-rich A≈ 120 mass region.