  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.
