Functional ZnO-nanowire/polymer core-shell heterostructures were realized using oxidative chemical vapour deposition (oCVD). This dry and versatile technique allows uniform coating of semiconductor nanowires with polymers and simultaneous doping control of the shell. Here, 100 nm thick, p-doped shells of poly(3,4-ethylenedioxythiophene) (PEDOT) were deposited around n-conductive ZnO nanowires. Energy-dispersive x-ray spectroscopy confirms the incorporation of Br dopants into the PEDOT shell, and the resulting p-conductivity of the polymer shell is demonstrated by electrical measurements on nanowire arrays. Photoluminescence spectroscopy points to reactions of Br with the ZnO surface but proves that the nanowires show only little degradation of their optical properties.