Maybe the answer to my question is obvious.


Let $p$ be a prime $\geq 3$. Let $D$ be an étale $(\varphi, \Gamma)$-module over $A\_{\mathbb{Q}\_p} = \{ \sum\_{n \in \mathbb{Z}} a\_n X^n \, \vert \, a\_n \in \mathbb{Z}\_p, a\_n \to 0 \mbox{ as } n\to -\infty \}$ of finite rank.


Recall that the action of $\varphi$ is given by $\varphi(X) = (1+X)^p -1$ and that of $\gamma \in \Gamma$ by $\gamma(X) = (1+X)^{\omega(\gamma)} -1$ where $\omega$ is the $p$-adic cyclotomic character.


Denote by $A^+$ the subring of $A\_{\mathbb{Q}\_p}$ of power series without denominators, that is $A^+ = \{ \sum\_{n \in \mathbb{N}} a\_n X^n \, \vert \, a\_n \in \mathbb{Z}\_p \}$.


Say that $D$ is of finite height if $D= D^+ \otimes\_{A^+} A\_{\mathbb{Q}\_p}$ where $D^+$ is an étale $(\varphi, \Gamma)$-module over $A^+$.


Is there an étale $(\varphi, \Gamma)$-module $D$ over $A\_{\mathbb{Q}\_p}$ (of finite rank) which is not of finite height and such that for all étale $(\varphi, \Gamma)$-module of rank $1$ $\delta$ over $A\_{\mathbb{Q}\_p}$, $D \otimes \delta$ is not of finite height ?


Obviously if such an object exists, it should be of rank $\geq 2$.



$(\varphi, \Gamma)$-modules of finite height
*Proposition*: If $D$ is not of finite height, then nor is $D \otimes \delta$ for any $\delta$ of rank 1. 


*Proof*: It suffices to show the contrapositive: if $D$ is of finite height so is $D \otimes \delta$ for any rank 1 $\delta$. This follows easily from the fact that every continuous character of $G\_{\mathbf{Q}\_p}$ (thus any etale rank 1 $(\varphi, \Gamma)$-module) is the product of an unramified character and a character factoring through $\Gamma$; twisting by a character of either type will just change the actions of $\varphi$ and of a generator of $\Gamma$ on $D$ by elements of $\mathbf{Z}\_p^\times$, and thus sends a $(\varphi, \Gamma)$-stable $\mathbf{A}^+\_{\mathbf{Q}\_p}$-sublattice in $D$ to one in $D(\delta)$.


Since representations that aren't of finite height do exist, the answer to your question is thus "yes" -- any non-finite-height $D$ will do.


