  We investigate the superconducting ternary lithium borohydride phase diagram
at pressures of 0 and 200$\,$GPa using methods for evolutionary crystal
structure prediction and linear-response calculations for the electron-phonon
coupling. Our calculations show that the ground state phase at ambient
pressure, LiBH$_4$, stays in the $Pnma$ space group and remains a wide band-gap
insulator at all pressures investigated. Other phases along the 1:1:$x$ Li:B:H
line are also insulating. However, a full search of the ternary phase diagram
at 200$\,$GPa revealed a metallic Li$_2$BH$_6$ phase, which is
thermodynamically stable down to 100$\,$GPa. This {\em superhydride} phase,
crystallizing in a $Fm\bar{3}m$ space group, is characterized by six-fold
hydrogen-coordinated boron atoms occupying the $fcc$ sites of the unit cell.
Due to strong hydrogen-boron bonding this phase displays a critical temperature
of $\sim$ 100$\,$K between 100 and 200$\,$GPa. Our investigations confirm that
ternary compounds used in hydrogen-storage applications are a suitable choice
for observing high-$T_\text{c}$ conventional superconductivity in diamond anvil
cell experiments, and suggest a viable route to optimize the critical
temperature of high-pressure hydrides.
