Zinc-blende boron arsenide (BAs) has been confirmed to have impressively high thermal conductivity. However, studies on its phase transitions under pressure have been few. Here, through recently developed structure search method, we predicted that many polytypes with structural features of cubic- and hexagonal-diamond, which are known to be unstable even up to very high pressure for carbon and boron nitride, can become stable at low pressure and might be retained to ambient conditions. Moreover, some of these BAs polytypes are calculated to have impressively high thermal conductivity both at ambient and high pressure, which opens up a new route to search for high thermal conductors.