The Th$_{7}$Fe$_{3}$ family of superconductors provides a rich playground for unconventional superconductivity. La$_7$Ni$_3$ is the latest member of this family, which we here investigate by means of thermodynamic and muon spin rotation and relaxation measurements. Our specific heat data provides evidence for two distinct and approximately isotropic superconducting gaps. The larger gap has a value slightly higher than that of weak-coupling BCS theory, indicating the presence of significant correlations. These observations are confirmed by transverse-field muon-rotation measurements. Furthermore, zero-field measurements reveal small internal fields in the superconducting state, which occur close to the onset of superconductivity and indicate that the superconducting order parameter breaks time-reversal symmetry. We discuss two possible microscopic scenarios -- an unconventional $E_{2}(1,i)$ state and an $s+i,s$ superconductor, which is reached by two consecutive transitions -- and illustrate which interactions will favor these phases. Our results establish La$_{7}$Ni$_{3}$ as the first member of the Th$_{7}$Fe$_{3}$ family displaying both time-reversal-symmetry-breaking and multigap superconductivity.
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