Let $\Omega$ be a completion of an algebraic closure of $\mathbb F_q\left(\left(\frac1T\right)\right)$ for the valuation $-\deg$. Consider two matrices $M_1,M_2$ in $\mathcal M_2(\Omega)$ that are $\Omega$-linearly independent. Denote by $\varphi$ the morphism of algebraic varieties $\varphi:\Omega^2\to\Omega^2$ defined by $\varphi\begin{pmatrix}z_1\\z_2\end{pmatrix}=M_1\begin{pmatrix}z_1\\z_2\end{pmatrix}+M_2\begin{pmatrix}z^q_1\\z^q_2\end{pmatrix}$. Does $\varphi(\Omega^2)$ contain a non empty open disk (for the topology induced by $\deg$) centered at $(0,0)$?

Let $\Omega$ be a completion of an algebraic closure of $\mathbb F_q\left(\left(\frac1T\right)\right)$ for the valuation $-\deg$. Consider two matrices $M_1,M_2$ in $\mathcal M_2(\Omega)$ that are $\Omega$-linearly independent. Denote by $\varphi$ the morphism of affine algebraic varieties $\varphi:\Omega^2\to\Omega^2$ defined by $\varphi\begin{pmatrix}z_1\\z_2\end{pmatrix}=M_1\begin{pmatrix}z_1\\z_2\end{pmatrix}+M_2\begin{pmatrix}z^q_1\\z^q_2\end{pmatrix}$. Does $\varphi(\Omega^2)$ contain a non empty open disk (for the topology induced by $\deg$) centered at $(0,0)$?