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The theory of $(\varphi_q,\Gamma)$-modules is a generalization of Fontaine'stheory of $(\varphi,\Gamma)$-modules, which classifies $G_F$-representations on$\CO_F$-modules and $F$-vector spaces for any finite extension $F$ of $\BQ_p$.In this paper following Colmez's method we classify triangulable$\CO_F$-analytic $(\varphi_q,\Gamma)$-modules of rank 2. In this process weestablish two kinds of cohomology theories for $\CO_F$-analytic$(\varphi_q,\Gamma)$-modules. Using them we show that, if $D$ is an$\CO_F$-analytic $(\varphi_q,\Gamma)$-module such that$D^{\varphi_q=1,\Gamma=1}=0$ i.e. $V^{G_F}=0$ where $V$ is the Galoisrepresentation attached to $D$, then any overconvergent extension of thetrivial representation of $G_F$ by $V$ is $\CO_F$-analytic. In particular,contrarily to the case of $F=\BQ_p$, there are representations of $G_F$ thatare not overconvergent.

Triangulable $\CO_F$-analytic $(\varphi_q,\Gamma)$-modules of rank 2