If the reaction shows a base is one of the reagents, that MEANS it is part of the rate determining step. The rate determining step is removal of the proton, hence it is an E2 reaction.
If you were to ask whether a tertiary halide could lose a halogen to the solvent without encountering the base, that could happen, but this would be a concentration dependent reaction competing with the E2 elimination. It is the concentrations that control the rates. It will be an E2 elimination. If concentrations were given, then it might, might be possible to argue that some of the reaction could have been SN1, but that is way beyond my pay grade.