Yes, it's correct. Written English tends to be more formal, though, and I would tend to write has been, but say it's. (It's not possessive, though. It's just a contraction for has, as it's is for it is.)
The New York Times is a well respected newspaper, which is fairly conservative in their English. Plus, it's written, which is more formal than spoken. However, in looking through some of their articles, I found:
- What if It's All Been a Big Fat Lie? - New York Times
- So Long, It's Been Good to Know You - NYTimes.com
- It's been a year since Citi Bike started in New York City... NYTimes.com
But even there, it tends to appear in blogs, in quoting someone, or in well known phrases (it's been good to know you.)