Using global structure searches, we have explored the structural stability of CaB3N3, a compound analogous to CaC6, under pressure. There are two high-pressure phases with space groups R3c and Amm2 that were found to be stable between 29 and 42 GPa, and above 42 GPa, respectively. The two phases show different structural frameworks, analogous to graphitic CaC6. Phonon calculations confirm that both structures are also dynamically stable at high pressures. The electronic structure calculations show that the R3c phase is a semiconductor with a band gap of 2.21 eV and that the Amm2 phase is a semimetal. These findings help advance our understanding of the Ca-B-N ternary system.