  Measurements of temperature-dependent resistance and magnetization under
hydrostatic pressures up to 2.13 GPa are reported for single-crystalline,
superconducting BaBi$_3$. A temperature - pressure phase diagram is determined
and the results suggest three different superconducting phases ${\alpha}$,
$\beta$, and $\gamma$ in the studied pressure range. We further show that
occurrence of the three superconducting phases is intuitively linked to phase
transitions at higher temperature which are likely first order in nature.
$T_p$, which separates phase ${\alpha}$ from $\beta$ and $\gamma$, is
associated with an abrupt resistance change as pressure is increased from 0.27
GPa to 0.33 GPa. Above 0.33 GPa, an "S-shape" anomaly in the
temperature-dependent resistance curve, $T_\text S$, is observed and associated
with the transition between the $\beta$ and $\gamma$ phases. Further increasing
of pressure above 1.05 GPa suppresses this transition and BaBi$_3$ stays in
$\gamma$ phase over the whole investigated temperature range. These
high-temperature anomalies are likely related to structural degrees of freedom.
With the ${\alpha}$ phase being the ambient-pressure tetragonal structure
($P4/mmm$), our first-principle calculations suggest the $\beta$ phase to be
cubic structure ($Pm-3m$) and the $\gamma$ phase to be a distorted tetragonal
structure where the Bi atoms are moved out of the face-centered position.
Finally, an analysis of the evolution of the superconducting upper critical
field with pressure further confirms these transitions in the superconducting
state and suggests a possible change of band structure or a Lifshitz transition
near 1.54 GPa in $\gamma$ phase. Given the large atomic numbers of both Ba and
Bi, our results establish BaBi$_3$ as a good candidate for the study of the
interplay of structure with superconductivity in the presence of strong
spin-orbit coupling.
