One gets loads of crystalline counterexamples using $p$-divisible groups (and more specifically from any elliptic curve with supersingular reduction).

Let $k$ be a perfect field of characteristic $p > 0$ and let $K$ be a complete discretely-valued field of characteristic 0 having residue field $k$. Let $\Gamma_0$ be a $p$-divisible group over $k$. By the unobstructedness of the infinitesimal deformation theory of $p$-divisible groups, this lifts to a $p$-divisible group $\Gamma$ over the valuation ring $O_K$ of $K$ (and even over $W(k)$). Suppose that the representation of the Galois group of $K$ associated to the generic fiber $\Gamma_K$ becomes reducible on the finite-index open subgroup, say associated to a finite extension $K'/K$. This is exactly the condition that $\Gamma_{K'}$ admits a nontrivial filtration, and by the usual schematic closure trick (and the work of Raynaud/Tate on $p$-divisible groups) that uniquely extends to such a filtration of $\Gamma_{O_{K'}}$, and hence of $(\Gamma_0)_{k'}$ for the residue field $k'/k$ of $K'/K$. In other words, as long as $\Gamma_0$ is a "geometrically isosimple" $p$-divisible group (i.e., $(\Gamma_0)_{\overline{k}}$ is simple in the isogeny category over $\overline{k}$) then all such $\Gamma_K$ furnish examples.

The Dieudonne-Manin classification provides lots of concrete Dieudonne modules corresponding to geometrically isosimple $\Gamma_0$ (and then by the Fontaine-Honda formalism or other formalisms one can "write down" the data of lots of lifts over $W(k)$). And in more concrete terms, the $p$-divisible group of any supersingular elliptic curve over $k$ is of this type (so the $p$-adic Tate module of any elliptic curve over $K$ with supersingular reduction does the job) since a nontrivial composition series would (by height reasons) have to consist of terms which are either multiplicative or etale, contradicting the supersingularity hypothesis.

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