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Abstract: Epsilon Indi A b is a directly imaged $\sim6 M_{\rm Jup}$ exoplanet orbiting a nearby (3.6 pc) K-dwarf at $\sim 30$ AU. We analyze archival JWST/MIRI 15 $μ$m coronagraphic imaging of this planet to search for directly imaged satellites orbiting Eps Ind A b. Within the planet's Hill sphere (radius $R_H \approx 2.3$ AU or $1.3 λ/D$), we compare single- and double-PSF models using Bayesian evidence. We find that a double-PSF (binary planet) fit is preferred. This apparent preference can most plausibly be explained by systematics, although follow-up observations would be required to fully rule out a binary planet interpretation. We construct a contrast curve of the exoplanet after removing this feature, demonstrating sensitivity to companions as faint as $0.03\times$ the F1550C flux of Eps Ind A b (equivalent to $T = 130$ K, $1.3 M_{\rm Jup}$) at large separations (>2 AU). We also demonstrate sensitivity to brighter companions $0.2\times$ the F1550C flux of Eps Ind A b (equivalent to $T = 180$ K, $2.5 M_{\rm Jup}$) down to separations of 0.52 AU (1.3 pixels; $0.29 λ/D$; 144 mas). This study demonstrates that JWST/MIRI can directly detect exomoons or binary planets inside the Hill sphere of directly imaged exoplanets orbiting neighboring stars.