The layered structure of superconducting cuprates is considered to be a key ingredient to achieve high superconducting transition temperatures. In this work, we investigate the possibility of doping the SrCuO₂ infinite-layer compound by inserting additional oxygen into its structure. We observe that the infinite-layer SrCuO₂ structure is epitaxially stabilized in thin films grown by pulsed laser deposition in pure O₂. Increasing the oxidizing power by introducing ozone during the growth leads to a different phase with an elongated c axis. Scanning transmission electron microscopy analysis suggests that the films with an elongated c axis are composed of SrCuO_2.5 blocks separated by SrCuO₂ layers arranged to match the substrate spacing. X-ray absorption spectroscopy measurements show that this SrCuO_{2+ δ} phase is associated with a more isotropic Cu orbital configuration and hole doping. This hole doping leads to a dramatic reduction in the resistivity of the films, with a magnitude that depends on the precise oxygen content in the structure.