Recently experimentalists have discovered several charged charmonium-like hadrons Z + c with unconventional quark content cc du. We perform a search for Z + c with mass below 4.2 GeV in the channel I G (J P C ) = 1 + (1 +− ) using lattice QCD. The major challenge is presented by the twomeson states J/ψ π, ψ2Sπ, ψ1Dπ, D D * , D * D * , ηcρ that are inevitably present in this channel. The spectrum of eigenstates is extracted using a number of meson-meson and diquark-antidiquark interpolating fields. For our pion mass of 266 MeV we find all the expected two-meson states but no additional candidate for Z + c below 4.2 GeV. Possible reasons for not seeing an additional eigenstate related to Z + c are discussed. We also illustrate how a simulation incorporating interpolators with a structure resembling low-lying two-mesons states seems to render a Z + c candidate, which is however not robust after further two-meson states around 4.2 GeV are implemented.