Are there any atheists in the Bible? Do any people in the Bible self identify as atheists?

Or are they all theists, differing in which god they follow?

Apparently at least some atheists existed, since David had heard of them:


  The fool says in his heart, “There is no God.” 


Psalm 14:1 and Psalm 53:1.
Not exactly the answer you are looking for, but Paul clearly understands humanity to be functionally atheistic, if not in practice. From Romans 1:


  18 The wrath of God is being revealed from heaven against all the godlessness and wickedness of people, who suppress the truth by their wickedness, 19 since what may be known about God is plain to them, because God has made it plain to them. 20 For since the creation of the world God’s invisible qualities—his eternal power and divine nature—have been clearly seen, being understood from what has been made, so that people are without excuse. 21 For although they knew God, they neither glorified him as God nor gave thanks to him, but their thinking became futile and their foolish hearts were darkened. 22 Although they claimed to be wise, they became fools 23 and exchanged the glory of the immortal God for images made to look like a mortal human being and birds and animals and reptiles.
   24 Therefore God gave them over in the sinful desires of their hearts to sexual impurity for the degrading of their bodies with one another. 25 They exchanged the truth about God for a lie, and worshiped and served created things rather than the Creator—who is forever praised. Amen.


In practice, to be a-theistic in the ancient world was to invite destruction upon the whole community. In Rome, in particular, for example, legally they didn't care which god you worshipped, as long as you worshipped one. I don't have the quote handy, but I remember David Bell in Many Mansions making the case that atheism was like "poor hygenie or a public health hazard," meaning that impiety would invite needless retribution from someone, so if everybody was worshipping what they thought, at the least the "true gods," would forgive them.