Despite the name, it is just one missile per round. 

[Deflect Missiles][1] says:
*"you can use your **reaction** to deflect or catch **the** missile when you are hit by **a** ranged weapon attack."* (the rest of the description similarly refers to just one attack). And since you only get one reaction per round, you can only deflect one attack.


  [1]: http://www.5esrd.com/classes/monk/#Deflect_Missiles