As a post-doc applicant, should I address a professor by their first name in email after they addressed me that way? Regarding a post-doc position, I recently wrote to a professor in the US by starting my mail "Dear Professor X" where X is the surname and she replied to me "Hello Y" where Y is my first name. 
I am hesitating if I should start always with "Dear Professor Y1" where Y1 is her first name. 
I did not find any satisfactory answer by searching here. Personally, I would choose the second option, starting by her first name but I am not sure if there is any social norm for that? What is the best appropriate way?
Continue to use the surname until explicitly given permission to do otherwise. One way for her to give that permission, actually, would be to sign a mail with only her first name. 

There is a power imbalance between you. She is probably not being entirely correct in assuming she can use your first name, but it is common for people in authority to take some liberties. 

But I'd recommend keeping it a bit formal until that obviously no longer works. And my full name is "Bobby Buffy" then I'd never be Professor Bobby. It would either be Professor Buffy or just Bobby. (For the record, though, I'm not Bobby.)



Note that this answer may not apply everywhere. It has a US perspective.