  For a smooth curve $B$ over an algebraically closed field $k$, for every
$B$-flat complete intersection $X_B$ in $B\times_{\text{Spec}\ k}
\mathbb{P}^n_k$ of type $(d_1,\dots,d_c)$, if the Fano index is $\geq 2$ and if
$\text{char}(k)>\max(d_1,\dots,d_c)$, we prove weak approximation of
$\widehat{\mathcal{O}}_{B,b}$-points of $X_B$ by $k(B)$-points at all places of
(strong) potentially good reduction, including all places of good reduction.
The key step is the proof that such complete intersections are \emph{separably
uniruled by lines}, and even \emph{separably rationally connected}, whenever
smooth. We prove that the inequality is close to sharp. We prove a similar
theorem for Fano manifolds of Picard number $1$ and Fano index $1$.

Algebraic Geometry