We study the production of the exotic charged charmoniumlike state Z±c(3900) in p¯p collisions through the sequential process ψ(4260)→Z±c(3900)π∓, Z±c(3900)→J/ψπ±. Using the subsample of candidates originating from semi-inclusive weak decays of b-flavored hadrons, we measure the invariant mass and natural width to be M=3902.6+5.2−5.0(stat)+3.3−1.4(syst)  MeV and Γ=32+28−21(stat)+26−7(syst)  MeV, respectively. We search for prompt production of the Z±c(3900) through the same sequential process. No significant signal is observed, and we set an upper limit of 0.70 at the 95% credibility level on the ratio of prompt production to the production via b-hadron decays. The study is based on 10.4  fb−1 of p¯p collision data collected by the D0 experiment at the Fermilab Tevatron collider.