  The $\phi^4$ model is coupled to an impurity in a way that preserves one-half
of the BPS property. This means that the antikink-impurity bound state is still
a BPS solution, i.e., a zero-pressure solution saturating the topological
energy bound. The kink-impurity bound state, on the other hand, does not
saturate the bound, in general. We found that, although the impurity breaks
translational invariance, it is, in some sense, restored in the BPS sector
where the energy of the antikink-impurity solution does not depend on their
mutual distance. This is reflected in the existence of a generalised
translational symmetry and a zero mode. We also investigate scattering
processes. In particular, we compare the antikink-impurity interaction close to
the BPS regime, which presents a rather smooth, elastic like nature, with other
scattering processes. However, even in this case, after exciting a sufficiently
large linear mode on the incoming antikink, we can depart from the close-to-BPS
regime. This results, for example, in a backward scattering.
