  Fast radio bursts (FRBs) are mysterious transient phenomena. The study of
repeating FRBs may provide useful information about their nature due to their
redetectability. The two most famous repeating sources are FRBs 121102 and
180916, with a period of 157 days and 16.35 days, respectively. Previous
studies suggest that the periodicity of FRBs is likely associated with neutron
star (NS) binary systems. Here we introduce a new model which proposes that
periodic repeating FRBs are due to the interaction of a NS with its planet in a
highly elliptical orbit. The periastron of the planet is very close to the NS
so that it would be partially disrupted by tidal force every time it passes
through the periastron. Fragments generated in the process could interact with
the compact star through the Alfv\'{e}n wing mechanism and produce FRBs. The
model can naturally explain the repeatability of FRBs with a period ranging
from a few days to several hundred days, but it generally requires that the
eccentricity of the planet's orbit should be large enough. Taking FRBs 121102
and 180916 as examples, it is shown that the main features of the observed
repeating behaviors can be satisfactorily accounted for.
