The low-temperature magnetic behavior of the double-layered ruthenate Sr₃Ru₂O₇, as grown from a eutectic Sr₂RuO₄-Sr₃Ru₂O₇ system, was investigated via zero- and transverse-field muon-spin rotation. The gradual increase of the muon relaxation rate observed below 2.5 K, even in the absence of applied magnetic fields, indicates the occurrence of a spontaneous breaking of time-reversal symmetry. The onset of the latter at a temperature above 1.5 K, the T_c of the single phase Sr₂RuO₄, provides evidence about an unconventional superconducting state in the eutectic phase, which most likely takes place at the interface between the Sr₂RuO₄ and Sr₃Ru₂O₇ domains, or even inside the Sr₃Ru₂O₇ phase. We show that the superconducting state manifests a two-component behavior in the transverse-field response with change-over at about T = 2.5 K and T = 1.5 K. The comparison with zero-field μSR data in the Ru-Sr₂RuO₄ eutectic system rules out the possibility of spurious effects due to embedded Ru islands.