  Taking advantage of a previously developed method, which allows to map
solutions of General Relativity into a broad family of theories of gravity
based on the Ricci tensor (Ricci-based gravities), we find new exact analytical
scalar field solutions by mapping the free-field static, spherically symmetric
solution of General Relativity (GR) into quadratic $f(R)$ gravity and the
Eddington-inspired Born-Infeld gravity. The obtained solutions have some
distinctive feature below the would-be Schwarzschild radius of a configuration
with the same mass, though in this case no horizon is present. The compact
objects found include wormholes, compact balls, shells of energy with no
interior, and a new kind of object which acts as a kind of wormhole membrane.
The latter object has Euclidean topology but connects antipodal points of its
surface by transferring particles and null rays across its interior in
virtually zero affine time. We point out the relevance of these results
regarding the existence of compact scalar field objects beyond General
Relativity that may effectively act as black hole mimickers.
