  The mixed spin-(1/2, 1) Ising model on two fully frustrated
triangles-in-triangles lattices is exactly solved with the help of the
generalized star-triangle transformation, which establishes a rigorous mapping
correspondence with the equivalent spin-1/2 Ising model on a triangular
lattice. It is shown that the mutual interplay between the spin frustration and
single-ion anisotropy gives rise to various spontaneously ordered and
disordered ground states, which differ mainly in an occurrence probability of
the non-magnetic spin state of the integer-valued decorating spins. We have
convincingly evidenced a possible coexistence of the spontaneous long-range
order with a partial disorder within the striking ordered-disordered ground
state, which manifest itself through a non-trivial criticality at finite
temperatures as well. A rather rich critical behaviour including the
order-from-disorder effect and reentrant phase transitions with either two or
three successive critical points is also found.
