Dust grains absorb half of the radiation emitted by stars throughout the history of the universe, re-emitting this energy at infrared wavelengths(1-3). Polycyclic aromatic hydrocarbons (PAHs) are large organic molecules that trace millimetre-size dust grains and regulate the cooling of interstellar gas within galaxies(4,5). Observations of PAH features in very distant galaxies have been difficult owing to the limited sensitivity and wavelength coverage of previous infrared telescopes(6,7). Here we present James Webb Space Telescope observations that detect the 3.3 mu m PAH feature in a galaxy observed less than 1.5 billion years after the Big Bang. The high equivalent width of the PAH feature indicates that star formation, rather than black hole accretion, dominates infrared emission throughout the galaxy. The light from PAH molecules, hot dust and large dust grains and stars are spatially distinct from one another, leading to order-of-magnitude variations in PAH equivalent width and ratio of PAH to total infrared luminosity across the galaxy. The spatial variations we observe suggest either a physical offset between PAHs and large dust grains or wide variations in the local ultraviolet radiation field. Our observations demonstrate that differences in emission from PAH molecules and large dust grains are a complex result of localized processes within early galaxies.