  By determining the superconducting lower and upper critical fields
$H_\mathrm{c1}(T)$ and $H_\mathrm{c2}(T)$, respectively, in a high-purity
spherical Sr$_2$RuO$_4$ sample via ac-susceptibility measurements, we obtain
the temperature dependence of the coherence length $\xi$ and the penetration
depth $\lambda$ down to 0.04$T_c$. Given the high sample quality, the observed
$T^2$ dependence of $\lambda$ at low temperatures cannot be explained in terms
of impurity effects. Instead, we argue that the weak type-II superconductor
Sr$_2$RuO$_4$ has to be treated in the non-local limit. In that limit, the
penetration depth data agree with a gap structure having vertical line nodes,
while horizontal line nodes cannot account for the observation.
