  We construct infinite families of irreducible supersingular mod $p$
representations of $\mathrm{GL}_2(F)$ with $\mathrm{GL}_2(\mathcal{O}_F)$-socle
compatible with Serre's modularity conjecture, where $F / \mathbb{Q}_p$ is any
finite extension with residue field $\mathbb{F}_{p^2}$ and ramification degree
$e \leq (p-1)/2$. These are the first such examples for ramified $F /
\mathbb{Q}_p$.
