Is the Birch and SwinnertonDyer conjecture known in positive characteristic?
Edit: This answer addresses an earlier version of the question, where the OP asked whether or not BSD made sense for elliptic curves over finite fields. It also however answers the current question.
The Birch and Swinnertondyer conjecture for an elliptic curve over a number field relates the rank of the MordellWeil group to the Lfunction of the curve. To have a "BSD" over other fields, one needs an analogue of these objects.
The appropriate positive characteristic analogue of BSD is for elliptic curves over function fields of curves (the other "global fields"). This is known in some cases, but not in full generality. In fact it is known that BSD for an elliptic curve over a function field is equivalent to the finiteness of the TateShafarevich group of the curve.
However, if you have an elliptic curve $E$ over a finite field $k$ and a function field $K$ of a curve in positive characteristic, then by base change you may consider $E_K$ as a "constant" curve over $K$. In which case I do believe that BSD is known, but hopefully someone else can help me with a precise reference as I cannot remember it.

5$\begingroup$ See e.g. Kato and Trihan, "On the conjectures of Birch and SwinnertonDyer in characteristic p > 0", math.umons.ac.be/ga/BSD.pdf. This proves finiteness of Sha implies BSD for general abelian varieties over function fields, and it gives references to lots of other results in the same vein, including the one you quote in your third paragraph (which is a theorem of Milne apparently). $\endgroup$ – David Loeffler Sep 18 '12 at 9:50

2$\begingroup$ people.math.gatech.edu/~ulmer/research/papers/2011.pdf by Ulmer contains a lot about BSD for elliptic curves in positive characteristic. $\endgroup$ – Chris Wuthrich Sep 18 '12 at 11:47

4$\begingroup$ The full BSD conjecture was stated for abelian varieties over all global fields by Tate in a 1966 Bourbaki talk. For elliptic curves over function fields, it's known that to prove the full BSD conjecture for an elliptic curve, it suffices to prove that some pcomponent of the Tate Shafarevich group is finite (MR0414558). In principle, for any specific curve, it is possible to check this condition. $\endgroup$ – anon Sep 18 '12 at 13:06