Laudal's Lemma in positive characteristic

Laudal’s Lemma states that if $C$ is a curve of degree $d>s^2+1$ in ${\mathbb{P}}^3$ over an algebraically closed field of characteristic 0 such that its plane section is contained in an irreducible curve of degree s, then $C$ lies on a surface of degree $s$. We show that the same result does not hold in positive characteristic and we find different bounds $d>f(s)$ which ensure that $C$ is contained in a surface of degree $s$.