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? 



Can an etale (phi, Gamma) module be an extension of non-etale ones?
In the first two cases, the slopes of $\varphi$-modules are given by the "standard" Dieudonné-Manin decomposition. In particular, subobjects of étale objects are étale.


For more info, see (for example) chapter 4.5 of Kedlaya's "Slope Filtrations Revisited".


