  Dust grains absorb half of the radiation emitted by stars throughout the
history of the universe, re-emitting this energy at infrared wavelengths.
Polycyclic aromatic hydrocarbons (PAHs) are large organic molecules that trace
millimeter-size dust grains and regulate the cooling of the interstellar gas
within galaxies. Observations of PAH features in very distant galaxies have
been difficult due to the limited sensitivity and wavelength coverage of
previous infrared telescopes. Here we present JWST observations that detect the
3.3um 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 the infrared emission
throughout the galaxy. The light from PAH molecules, large dust grains, and
stars and hot dust are spatially distinct from one another, leading to
order-of-magnitude variations in the PAH equivalent width and the ratio of PAH
to total infrared luminosity across the galaxy. The spatial variations we
observe suggest either a physical offset between the PAHs and large dust grains
or wide variations in the local ultraviolet radiation field. Our observations
demonstrate that differences in the emission from PAH molecules and large dust
grains are a complex result of localized processes within early galaxies.
