  To explore superfluidity in flat-band systems, we consider a Bose-Hubbard
model on a cross-linked ladder with $\pi$ flux, which has a flat band with a
gap between the other band for noninteracting particles, where we study the
effect of the on-site repulsion nonperturbatively. For low densities, we find
exact degenerate ground states, each of which is a Wigner solid with
nonoverlapping Wannier states on the flat band. At higher densities, the
many-body system, when projected onto the lower flat band, can be mapped to a
spin-chain model. This mapping enables us to reveal the existence of a
Tomonaga-Luttinger liquid comprising it pairs of bosons. Interestingly, the
high- and low- density regions have an overlap, where the two phases coexist.
