  The nontrivial topology of {\it p}-wave superfluids make these systems
attractive candidate in information technology. In this work we report on the
topological state of a {\it p}-wave nonequilibrium excitonic insulator (NEQ-EI)
and show how to steer a nontopological band-insulator with bright $p$-excitons
toward this state by a suitable laser pulse, thus achieving a dynamical
topological phase transition. The underlying mechanism behind the transition is
the broken gauge-symmetry of the NEQ-EI which causes self-sustained persistent
oscillations of the excitonic condensate and hence a Floquet topological state
for high enough exciton densities. We show the formation of Floquet Majorana
modes at the boundaries of the open system and discuss topological spectral
signatures for ARPES experiments. We emphasize that the topological properties
of a {\it p}-wave NEQ-EI arise exclusively from the electron-hole Coulomb
interaction as the system is not driven by external fields.
