Let $K$ be a field of characteristic $p\>0$, $C$ a proper, smooth, geometrically connected curve over $K$, and 0 and $\infty$ two $K$-rational points on $C$. We show that any representation of the local Galois group at $\infty$ extends to a representation of the fundamental group of $C-\{0,\infty \}$ which is tamely ramified at 0, provided either that $K$ is separately closed or that $C$ is ${\mathbf{P}}^1$. In the latter case, we show there exists a unique such extension, called “canonical”, with the property that the image of the geometric fundamental group...