The cross section for ee+ e- → π+ π- J/ψ between 3.8 and 5.5 GeV is measured with a 967 fb(-1) data sample collected by the Belle detector at or near the Υ(nS) (n = 1,2,…,5) resonances. The Y(4260) state is observed, and its resonance parameters are determined. In addition, an excess of π+ π- J/ψ production around 4 GeV is observed. This feature can be described by a Breit-Wigner parametrization with properties that are consistent with the Y(4008) state that was previously reported by Belle. In a study of Y(4260) → π+ π- J/ψ decays, a structure is observed in the M(π(±)J/ψ) mass spectrum with 5.2σ significance, with mass M = (3894.5 ± 6.6 ± 4.5) MeV/c2 and width Γ = (63 ± 24 ± 26) MeV/c2, where the errors are statistical and systematic, respectively. This structure can be interpreted as a new charged charmoniumlike state.