Let $G = \mathbb Z / 3\mathbb Z$ be the cyclic group of order three. I conjecture that every irreducible representation over a finite field of characteristic $2$ must be trivial. But how to prove this?
Note, this does not hold if the field is infinite: If $K$ is the algebraic closure of the field $\mathbb F_2 = \{0,1\}$ (and hence infinite), then I know that every irreducible representation over $K$ must have dimension one as $G$ is abelian, hence we could take $K$ as a vector space over itself in our search for representations. Hence $G$ has to be embedded into the multiplicative group of $K$. Denote by $\rho : G \to K$ such an embedding, if $\rho$ is not the trivial representation we have $\rho(g) \ne 1$ and $\rho(g)^3 = 1$, hence $\rho(g)$ is a nontrivial root of the equation $x^3 - 1$ and we can choose two distinct values for $\rho(g)$, giving rise to two different representations; hence in this case we have three representations.