  We formulate and prove an analogue of the non-commutative Iwasawa Main
Conjecture for $\ell$-adic representations of the Galois group of a function
field of characteristic $p$. We also prove a functional equation for the
resulting non-commutative $L$-functions. As corollaries, we obtain
non-commutative generalisations of the main conjecture for Picard-$1$-motives
of Greither and Popescu and a main conjecture for abelian varieties over
function fields in precise analogy to the $\operatorname{GL}_2$ main conjecture
of Coates, Fukaya, Kato, Sujatha and Venjakob.
