CR7 is among the most luminous Ly α emitters (LAEs) known at z = 6.6 and consists of at least three UV components that are surrounded by Ly α emission. Previous studies have suggested that it may host an extreme ionizing source. Here, we present deep integral field spectroscopy of CR7 with VLT/Multi Unit Spectroscopic Explorer (MUSE). We measure extended emission with a similar halo scale length as typical LAEs at z ≍ 5. CR7's Ly α halo is clearly elongated along the direction connecting the multiple components, likely tracing the underlying gas distribution. The Ly α emission originates almost exclusively from the brightest UV component, but we also identify a faint kinematically distinct Ly α emitting region nearby a fainter component. Combined with new near-infrared data, the MUSE data show that the rest-frame Ly α equivalent width (EW) is ≍100 Å. This is a factor 4 higher than the EW measured in low-redshift analogues with carefully matched Ly α profiles (and thus arguably H I column density), but this EW can plausibly be explained by star formation. Alternative scenarios requiring active galactic nucleus (AGN) powering are also disfavoured by the narrower and steeper Ly α spectrum and much smaller IR to UV ratio compared to obscured AGN in other Ly α blobs. CR7's Ly α emission, while extremely luminous, resembles the emission in more common LAEs at lower redshifts very well and is likely powered by a young metal-poor starburst.