MTV has some comments from Irrational's Ken Levine suggesting that the PS Vita is an all-new construct. While Bioshock Infinite wowed everyone at E3, I'd be happy with another dose of sub-aquatic city spelunking and it'll definitely be a core game, not just the mini-games or some other mad idea, yay!
So, we'll see an entirely new "BioShock" released on the Vita platform. Levine also went on to clarify that the handheld game isn't a tie-in to "Infinite," and was "being built from the ground up." That said, it also doesn't sound like a port of the first game has been entirely ruled out. Levine seems to insulate that a portable "BioShock 1," would likely be ported outside of Irrational – a common practice when it comes to re-releases.