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 the last category isn't closed under extensions: it often happens that the Robba-ring $(\varphi, \Gamma)$-module of a representation can be written as an extension of other Robba-ring $(\varphi, \Gamma)$-modules which are not themselves of slope 0, and there is the whole rich theory of trianguline representations.

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?