We consider the difference equation $\sum_{j=1}^q a_j(x)\varphi(x+\alpha_j)=\chi(x)$ where $\alpha_1<\dots<\alpha_q$ ($q\geq 3$) are given real constants, $a_j$ ($j=1,\dots,q$) are given holomorphic functions on a strip $\mathbb{R}_{\delta }$ ($\delta>0$) such that $a_1$ and $a_q$ vanish nowhere on it, and $\chi$ is a function belonging to a quasianalytic Carleman class $C_M\{\mathbb{R}\}$. We prove, under a growth condition on the functions $a_j$, that the difference equation above is solvable in $C_M\{\mathbb{R}\}$.