  We discuss a new extended gravity model in ordinary $D=4$ spacetime
dimensions, where an additional term in the action involving Gauss-Bonnet
topological density is included without the need to couple it to matter fields
unlike the case of ordinary D=4 Gauss-Bonnet gravity models. Avoiding the
Gauss-Bonnet density becoming a total derivative is achieved by employing the
formalism of metric-independent non-Riemannian spacetime volume-forms. The
non-Riemannian volume element triggers dynamically the Gauss-Bonnet scalar to
become an arbitrary integration constant on-shell. We describe in some detail
the class of static spherically symmetric solutions of the above modified D=4
Gauss-Bonnet gravity including solutions with deformed (anti)-de Sitter
geometries, black holes, domain walls and Kantowski-Sachs-type universes. Some
solutions exhibit physical spacetime singular surfaces not hidden behind
horizons and bordering whole forbidden regions of space. Singularities can be
avoided by pairwise matching of two solutions along appropriate domain walls.
For a broad class of solutions the corresponding matter source is shown to be a
special form of nonlinear electrodynamics whose Lagrangian L(F^2) is a
non-analytic function of F^2 (the square of Maxwell tensor F_mu_nu), i.e.,
L(F^2) is not of Born-Infeld type.
