Summary: We start by revisiting the problem of finding BPS solutions in \(\mathcal{N} = 4\) \(\mathrm{SU}(2)\times\mathrm{SU}(2)\) gauged supergravity. We report on a new supersymmetric solution in the abelian sector of the theory, which describes a soliton that is regular everywhere. The solution is 1/4 BPS and can be obtained from a double analytic continuation of a planar solution found by D. Klemm [Nucl. Phys., B 545, No. 1–3, 461–478 (1999; Zbl 0953.83019)]. Also in the abelian sector, but now for a spherically symmetric ansatz we find a new solution whose supersymmetric nature was overlooked in the previous literature. Then, we move to the non-abelian sector of the theory by considering the meron ansatz for SU(2). We construct electric-meronic and double-meron solutions and show that the latter also leads to 1/4 BPS configurations that are singular and acquire an extra conformal Killing vector. We then move beyond the supergravity embedding of this theory by modifying the self-interaction of the scalar, but still within the same meron ansatz for a single gauge field, which is dilatonically coupled with the scalar. We construct exact black holes for two families of self-interactions that admit topologically Lifshitz black holes, as well as other black holes with interesting causal structures and asymptotic behavior. We analyze some thermal properties of these spacetimes.