No, Google's Street View car did not run over this donkey! Fortunately this donkey got away in one piece. In a post titled "Never Ass-ume," Kei Kawai, a Group Project Manager of Google Maps said "I’m pleased to confirm the donkey is alive and well" and published photographic evidence showing that the donkey got back up on it's feet and walked away.
So what was the donkey doing lying in the path of the Google Street View car? It was probably just enjoying a dust bath.