- 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.