This question is about p-adic representations of $\mathrm{Gal}(\overline{\mathbb{Q}}_p / \mathbb{Q}_p)$ and $(\varphi, \Gamma)$-modules. By theorems of Fontaine, Cherbonnier-Colmez and Kedlaya, the category of p-adic representations of $\mathrm{Gal}(\overline{\mathbb{Q}}_p / \mathbb{Q}_p)$ is equivalent to each of the following three categories:
- etale $(\varphi, \Gamma)$-modules over Fontaine's ring $\mathbb{B}_{\mathbb{Q}_p}$
- etale $(\varphi, \Gamma)$-modules over the subring $\mathbb{B}^{\dagger}_{\mathbb{Q}_p}$
- slope zero $(\varphi, \Gamma)$-modules over the Robba ring $\mathcal{R}$ (also known as $\mathbb{B}^{\dagger}_{\mathrm{rig}, \mathbb{Q}_p}$).
It's well known that slope 0 $(\varphi, \Gamma)$-modules over the Robba ring can sometimes be written as extensions of other Robba-ring $(\varphi, \Gamma)$-modules which are not themselves of slope 0. (Indeed there is the whole rich theory of trianguline representations, whose Robba-ring $(\varphi, \Gamma)$-modules are built up entirely from rank 1 pieces.)
My question: does this happen for either of the other two categories of $(\varphi, \Gamma)$-modules? Can one have a short exact sequence of $(\varphi, \Gamma)$-modules over $\mathbb{B}_{\mathbb{Q}_p}$ or $\mathbb{B}^{\dagger}_{\mathbb{Q}_p}$ where the middle term is etale but the two end terms are not?
