Actually, a lot of inscriptions from around the 3rd century AD are in Latin, Greek and Punic.
By that period, vernacular Punic had become Neo-Punic, which was basically Punic with some aspects of Berber. But it seems that what language was dominant depended on the particular context within pre-Islamic North Africa (etc. countryside, city, etc).
The fact that Arabic became dominant so quickly in North Africa implies that there were a lot of Neo-Punic speakers, since languages in the same language group tend to "crowd out" others quicker, so people whose parents and grandparents spoke Neo-Punic would use Arabic instead, which is relatively similar.