  Using the recent work of Frankland and Spitzweck, we define Steenrod
operations $P^{n}$ on the mod $p$ motivic cohomology of smooth varieties
defined over a base field of characteristic $p$. We show that $P^{n}$ is the
$p$th power on $H^{2n,n}(-,\mathbb{F}_{p})\cong CH^{n}(-)/p$ and prove an
instability result for the operations. Restricted to mod $p$ Chow groups, we
show that the operations satisfy the expected Adem relations and Cartan
formula. For $p=2$, we use the new Steenrod squares to obtain new results on
quadratic forms over a base field of characteristic $2$.
