We study the process ee+ e- → π+ π- J/ψ at a center-of-mass energy of 4.260 GeV using a 525 pb(-1) data sample collected with the BESIII detector operating at the Beijing Electron Positron Collider. The Born cross section is measured to be (62.9±1.9±3.7) pb, consistent with the production of the Y(4260). We observe a structure at around 3.9 GeV/c2 in the π(±)J/ψ mass spectrum, which we refer to as the Z(c)(3900). If interpreted as a new particle, it is unusual in that it carries an electric charge and couples to charmonium. A fit to the π(±)J/ψ invariant mass spectrum, neglecting interference, results in a mass of (3899.0±3.6±4.9) MeV/c2 and a width of (46±10±20) MeV. Its production ratio is measured to be R = (σ(e+ e- → π(±)Z(c)(3900)(∓) → π+ π- J/ψ)/σ(e+ e- → π+ π- J/ψ)) = (21.5±3.3±7.5)%. In all measurements the first errors are statistical and the second are systematic.