**Saruman didn't break the staff, because he managed to take it whole** from Gandalf and kept it (checkout [the end of the fight][1] where Saruman has both staffs). 

**When Gandalf flees Orthanc he does so without his staff** (revisit the scene [here][2]) and later on gets another staff. This differs in the films from the books. The accounts of how he gets another staff in the movie universe differ a bit and maybe this is worth a separate question, but he clearly looses his staff in the fight with Saruman.

Further somewhat authoritative sources for this answer are this [IMDB FAQ][3] (see How did Gandalf get his staff back from Saruman? He simply leaps off of Orthanc without it) and the [Lotr wiki][4].

The flight scene also shows that Saruman still hoped he could turn Gandalf till that moment, so it was reasonable to keep his staff intact, but out of reach of Gandalf. Also note that in the movie universe, wizards obviously can make use of the staffs of other wizards (perhaps less efficiently). So, even if he would not convince Gandalf it might be prudent to keep the staff just like you might keep a spare sword around, if you have it anyway. 



  [1]: https://youtu.be/p1Vyhve9gtg?t=191
  [2]: https://www.youtube.com/watch?v=kNnvcs-sQB8
  [3]: https://www.imdb.com/title/tt0120737/faq#.2.1.15
  [4]: http://lotr.wikia.com/wiki/Wizard_Staff