This question unfortunately has a very similar name to this one, but I what want to ask here is different.

Let $K$ be a finite extension of $\mathbb{Q}_p$. It seems to be well known that the local version of the Fontaine-Mazur conjecture is false, in the sense that if $\rho:Gal(\overline{K}/K)\rightarrow GL_d(\mathbb{Q}_p)$ is some de Rham representation, it may very well be that $\rho$ does not come from a variety defined over a $p$-adic field. For instance, this is claimed by Mark Kisin in his survery paper "What is... a Galois representation?".

Maybe this question is naive, but can anyone give me a concrete example of such a representation?

Also, a slightly unrelated question: does one expect the answer to this version of the local Fontaine-Mazur conjecture to change if we allow representations coming from the etale cohomology of **analytic** spaces?

1more comment