  The hitherto unprecedented angular resolution of the Event Horizon Telescope
(EHT) has created exciting opportunities in the search for new physics.
Recently, the linear polarization of radiation emitted near the supermassive
black hole M87$^\star$ was measured on four separate days, precisely enabling
tests of the existence of a dense axion cloud produced by a spinning black
hole. The presence of an axion cloud leads to a frequency-independent
oscillation in the electric vector position angle (EVPA) of this linear
polarization. For a nearly face-on M87$^\star$, this oscillation in the EVPA
appears as a propagating wave along the photon ring. In this paper, we leverage
the azimuthal distribution of EVPA measured by the EHT to study the
axion-photon coupling. We propose a novel differential analysis procedure to
reduce the astrophysical background, and derive stringent constraints on the
existence of axions in the previously unexplored mass window $\sim
(10^{-21}-10^{-20})$~eV.
