To explore whether a flat-band system can accommodate superconductivity, we consider repulsively interacting fermions on the diamond chain, a simplest possible quasi-one-dimensional system that contains a flat band. Exact diagonalization and the density-matrix renormalization group are used to show that we have a significant binding energy of a Cooper pair with a long-tailed pair-pair correlation in real space when the total band filling is slightly below 1/3, where a filled dispersive band interacts with the flat band that is empty but close to E-F. Pairs selectively formed across the outer sites of the diamond chain are responsible for the pairing correlation. At exactly 1/3-filling an insulating phase emerges, where the entanglement spectrum indicates the particles on the outer sites are highly entangled and topological. These come from a peculiarity of the flat band in which "Wannier orbits" are not orthogonalizable.