"Sofia prefers to lie on her back."
Translation:Sofia preferas kuŝi sur sia dorso.
I have to say hearing it in Esperanto confused me a bit too, but come and think of it, it's the way it's said in a lot of European languages. Including English, German, French, Dutch, and Swedish.
On tekstaro.com (a searchable database of Esperanto texts by esteemed writers, including Zamenhof himself) I can find over a hundred examples of "sur la dorso," "sur mia dorso," "sur sia dorso," and so on, but the only instances with "per" are "kovrante per sia dorso" (covering with his back) and "apogita per la dorso" (supported by the back). So "sur" is definitely the preposition to use in Esperanto.
It does make some sense, though. When lying on your back, the rest of your body is lying on top of your back, in a sense. But I'll give it to you "per" would have been a little more logical to me as well.