Splitting properties of the reduction of semi-abelian varieties

Let $K$ be a complete discrete valuation field. Let ${\mathcal{O}}_K$ be its ring of integers. Let $k$ be its residue field which we assume to be algebraically closed of characteristic exponent $p\ge 1$. Let $G\mathrm{/}K$ be a semi-abelian variety. Let $\mathcal{G}\mathrm{/}{\mathcal{O}}_K$ be its N'eron model. The special fiber ${\mathcal{G}}_k\mathrm{/}k$ is an extension of the identity component ${\mathcal{G}}_k^0\mathrm{/}k$ by the group of components $\mathrm{\Phi }(G)$. We say that $G\mathrm{/}K$ has split reduction if this extension is split. Whereas $G\mathrm{/}K$ has always split reduction if $p=1$ we prove that it is no longer the case if $p>1$ even if $G\mathrm{/}K$ is tamely ramified. If $J\mathrm{/}K$ is the Jacobian variety of a smooth proper and geometrically connected curve $C\mathrm{/}K$ of genus $g$, we prove that for any tamely ramified extension $M\mathrm{/}K$ of degree greater than a constant, depending on $g$ only, $J_M\mathrm{/}M$ has split reduction. This answers some questions of Liu and Lorenzini.