  Let $F$ be a non-Archimedean local field with the residual characteristic
$p$. We construct a "good" number of smooth irreducible
$\bar{\mathbf{F}}_p$-representations of $GL_2(F)$, which are supersingular in
the sense of Barthel and Livn\'e. If $F=\mathbf{Q}_p$ then results of Breuil
imply that our construction gives all the supersingular representations up to
the twist by an unramified quasi-character. We conjecture this is true for
arbitrary $F$.
