  Period and index of a curve $X/K$ over a $p$-adic local field $K$ such that
the fundamental group $\pi_1(X/K)$ admits a splitting are shown to be powers of
$p$. As a consequence, examples of curves over number fields are constructed
where having sections is obstructed locally at a $p$-adic place. Hence the
section conjecture holds for these curves as there are neither sections nor
rational points.
