Splitting properties of the reduction of semi-abelian varieties

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