  We study phenomena happening at the surface of a star in Eddington-inspired
Born-Infeld (EiBI) gravity. The star is made of particles, which are
effectively described by a polytropic fluid. The EiBI theory was known to have
a pathology that singularities happen at a star surface. We suggest that the
gravitational back-reaction on the particles cures the problem. Strong tidal
forces near the (surface) singularity modify the effective equation of state of
the particles or make the surface be unstable depending on its matter contents.
The geodesic deviation equations take after the Hooke's law, where its
frequency-squared is proportional to the scalar curvature at the surface. For a
positive curvature, a particle collides with a probing wall more often and
increases the pressure. With the increased pressure, the surface is no longer
singular. For a negative curvature, the matters around the surface experience
repulsions with infinite accelerations. Therefore, the EiBI gravity is saved
from the pathology of surface singularity.
