Semiconductor nanowires of high purity and crystallinity hold promise as building blocks for miniaturized optoelectrical devices. Using scanning-excitation single-wire emission spectroscopy, with either a laser or an electron beam as a spatially resolved excitation source, we observe standing-wave exciton polaritons in ZnO nanowires at room temperature. The Rabi splitting between the polariton branches is more than 100 meV. The dispersion curve of the modes in the nanowire is substantially modified due to light-matter interaction. This finding forms a key aspect in understanding subwavelength guiding in these nanowires.