Spiral galaxies have distinct internal structures, including a stellar bulge, a disk, and spiral arms. It is unknown when in cosmic history these structures formed. In this study, we analyzed observations of BRI 1335-0417, an intensely star-forming galaxy in the distant Universe, at a redshift of 4.41. The [C II] gas kinematics shows a steep velocity rise near the galaxy center and has a two-armed spiral morphology, which extends from about 2 to 5 kiloparsecs in radius. We interpret these features as due to a central compact structure such as a bulge, a rotating gas disk, and either spiral arms or tidal tails. These features had formed within 1.4 billion years after the Big Bang, long before the peak of cosmic star formation.