Smaller, yes, but not small, and it isn't impossible to tame the Savanna Elephant. I've seen tamed Savanna Elephants on TV.
I think the Carthaginians used a few of them too. Anyway, it took great skill to effectively use elephants and it required a "culture" of elephant taming. India had this with their mahouts, estensive non-military use of elephants etc. My point is that North Africa could have had this.