  A massive body with the Schwarzschild interior metric (perfect fluid of
constant density) develops a pressure singularity at the origin when the radius
of the body $R$ approaches $9r_s/8$, where $r_s$ is the Schwarzschild radius.
We show that a quantum scalar particle scattered in this gravitational field
possesses a dense spectrum of narrow resonances. Their density and lifetimes
tend to infinity in the limit $R\rightarrow 9r_s/8$, and we determine the cross
section of the particle capture into these quasibound states. Therefore, a body
that is not a black hole demonstrates black-hole-like absorption.
