Non-admissible irreducible representations of $p$-adic ${\mathrm{G}\mathrm{L}}_n$ in characteristic $p$

Let $p>3$ and $F$ be a non-archimedean local field with residue field a proper finite extension of ${\mathbb{F}}_p$. We construct smooth absolutely irreducible non-admissible representations of ${\mathrm{G}\mathrm{L}}_2(F)$ defined over the residue field of $F$ extending the earlier results of the authors for $F$ unramified over ${\mathbb{Q}}_p$. This construction uses the theory of diagrams of Breuil and Paskunas. By parabolic induction, we obtain smooth absolutely irreducible non-admissible representations of ${\mathrm{G}\mathrm{L}}_n(F)$ for $n>2$.