  Given a symplectic surface $(\Sigma, \omega)$ of genus $g \ge 4$, we show
that the free group with two generators embeds into every asymptotic cone of
$(\mathrm{Ham}(\Sigma, \omega), d_\mathrm{H})$, where $d_\mathrm{H}$ is the
Hofer metric. The result stabilizes to products with symplectically aspherical
manifolds.
