  Spin-triplet chiral p-wave superconductivity is typically described by a
two-component order parameter, and as such is prone to unique emergent effects
when compared to the standard single-component superconductors. Here we present
the equilibrium phase diagram for small mesoscopic chiral p-wave
superconducting disks in the presence of magnetic field, obtained by solving
the microscopic Bogoliubov-de Gennes equations self-consistently. In the
ultra-small limit, the cylindrically-symmetric giant-vortex states are the
ground state of the system. However, with increasing sample size, the
cylindrical symmetry is broken as the two components of the order parameter
segregate into domains, and the number of fragmented domain walls between them
characterizes the resulting states. Such domain walls are topological defects
unique for the p-wave order, and constitute a dominant phase in the mesoscopic
regime. Moreover, we find two possible types of domain walls, identified by
their chirality-dependent interaction with the edge states.
