Although the possibility of spatial variations in the superfluid of unconventional, strongly correlated superconductors has been suggested1-7, it is not known whether such inhomogeneities-if they exist-are driven by disorder, strong scattering or other factors. Here we use atomic-resolution Josephson scanning tunnelling microscopy to reveal a strongly inhomogeneous superfluid in the iron-based superconductor FeTe0.55Se0.45. By simultaneously measuring the topographic and electronic properties of the superconductor, we find that this inhomogeneity in the superfluid is not caused by structural disorder or strong inter-pocket scattering and is not correlated with variations in the energy required to break electron pairs. Instead, we see a clear spatial correlation between the superfluid density and the quasiparticle strength (the height of the coherence peak) on a local scale. This result places iron-based superconductors on equal footing with copper oxide superconductors, where a similar relation has been observed on the macroscopic scale. Our results establish the existence of strongly inhomogeneous superfluids in unconventional superconductors, excluding chemical disorder and inter-band scattering as the causes of the inhomogeneity, and shed light on the relation between quasiparticle character and superfluid density. When repeated at different temperatures, our technique could further help to elucidate what local and global mechanisms limit the critical temperature in unconventional superconductors.