As you know the elephant is used in some parts of
>the world as a mode of transport and therefore it is possible to say
>that somebody 'travels by elephant' (we're not talking about the
>animal itself here).
Do you know what the asker needs these phrases for? Possible does not always mean appropriate.
Translate the sentences below for yourself and I hope you will understand my point.
Mam dżipa. Pojechałem dżipem do kina.
Mam fiata. Pojechałem fiatem do kina.
I am more inclined to accept "by elephant" as the setting that I imagine he uses in his school composition (?) would most probably by one connected with Africa, or India for that matter. Or perhaps he just wants to prove to his teacher that he knows better. In any case, it is the context that can tell us whether this usage is appropriate. What is my second guess holds and the teacher is you?