When did Baghdad Start?

Here's a nice little mystery.

According to the Columbia Encyclopedia (which I prefer to Wikepedia for reliability), Baghdad was founded by the Caliph Mansur, in 762. This was a while ago, but by the standards of the region it's actually almost modern; by the 8th century the recorded history of this part of the world already had a couple millennia under its belt.

In our Talmud study group we yesterday noted in passing the appearance in a duscussion of one Rav Hanna of Baghdat. One of our group, who knows a lot, commented that this is the first and only mention of Baghdad he's aware of in the Talmud, most of which took place in older Babylonian cities such as Sura, Pompedita, and Nahardea.

Except that the passage seems to be talking about the early Amoraitic period, say, third or fourth century at the latest, and in any case not the eighth.

Bava Batra 142b.