Is there an Anime in which one of the characters use a smartphone?

yes but its not an anime yet but some guy from kagerou project does have a smartphone and his name is Shintarou and it is still an manga though

Tomoko from Watamote has a smartphone. Here's the ending. The smartphone isn't exactly a big plot device though, she just uses it to call her one friend. I forget if she plays games in class with it or not or if she uses a handheld game system for that.