  Superconductors (SCs) with nontrivial topological band structures in the
normal state have been discovered recently in bulk materials. When such SCs are
made into thin films, quantum tunneling and Cooper pairing take place between
the topological surface states (TSSs) on the opposing surfaces. Here, we find
that chiral topological superconductivity with spontaneous time-reversal
symmetry breaking emerges on the surface of such thin film SCs. There is a
mirror symmetry that protects a novel nonunitary orbital and spin triplet
pairing of the TSS. In the mirror diagonal space, the chiral topological SC
manifests as two independent chiral $p$-wave spin-triplet pairing states, in
which each is a two-dimensional superconducting analog of the
Anderson-Brinkman-Morel state in superfluid $^3$He with in-plane exchange
fields. A rich topological phase diagram governed by the nontrivial
$\mathbb{Z}\oplus\mathbb{Z}$ topological invariant is obtained with gapless
chiral Majorana edge modes and anyonic Majorana vortices. We further construct
a three dimensional lattice model with a topological band structure and SC
pairings, which is motivated by Fe-based SCs such as Fe(Te,Se). We demonstrate
the realization of the proposed intrinsic chiral topological superconductor in
the quasi-two-dimensional thin-film limit. Our findings enable thin-film SCs
with nontrivial $Z_2$ band structures as a single-material platform for
intrinsic chiral topological superconductivity with both vortex and boundary
Majorana excitations for topological quantum device making.
