  We prove that any non-isotrivial elliptic K3 surface over an algebraically
closed field $k$ of arbitrary characteristic contains infinitely many rational
curves. In the case when $\mathrm{char}(k)\neq 2,3$, we prove this result for
any elliptic K3 surface. When the characteristic of $k$ is zero, this result is
due to the work of Bogomolov-Tschinkel and Hassett.
