  Gravitational scattering events, in which the path of an interstellar object
is deflected by a pulsar or the solar system, give rise to reflex motion which
can potentially be detected using pulsar timing. We determine the form of the
timing signal expected from a gravitational scattering event, which is
ramp-like and resembles the signal produced by a glitch or a gravitational wave
burst with memory (BWM), and investigate the prospects for detecting such a
signal using a pulsar timing array. The level of timing precision currently
achieved for some millisecond pulsars makes it possible to detect objects as
small as $10^{-10}$ solar masses, less than the mass of the dwarf planet Ceres,
at impact parameters as large as 1 AU. The signals produced by gravitational
scattering could provide independent constraints on models of dark matter
involving asteroid-mass objects or subhalos, and should be considered as
potential false positives in searches for BWMs.
