Question

Is the Birch and Swinnerton-Dyer conjecture known in positive characteristic?

Answer 1

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 Swinnerton-dyer conjecture for an elliptic curve over a number field relates the rank of the Mordell-Weil group to the L-function 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 Tate-Shafarevich 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.