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We discuss exact scalar field solutions describing gravitating compact objects in the Eddington-inspired Born-Infeld gravity (EiBI), a member of the class of (metric-affine formulated) Ricci-based gravity theories (RBGs). We include a detailed account of the RBGs/GR correspondence exploited to analytically solve the field equations. The single parameter $\e$ of the EiBI model defines two branches for the solution. The $\e>0$ branch may be described as a `shell with no interior', and constitutes an ill-defined, geodesically incomplete spacetime. The more interesting $\e<0$ branch admits the interpretation of a `wormhole membrane', an exotic horizonless compact object with the ability to transfer particles and light from any point on its surface (located slightly below the would-be Schwarzschild radius) to its antipodal point, in a vanishing fraction of proper time. This is a single example illustrating how the structural modifications introduced by the metric-affine formulation may lead to significant departures from GR even at astrophysically relevant scales, giving rise to physically plausible objects radically different from those we are used to think of in the metric approach, and that could act as a black hole mimickers whose shadows might present distinguishable signals.