Gelfand-Kirillov bound for $p$-adic Banach representations with infinitesimal character for ${\text{GL}}_2$ and quaternion units

We prove that an admissible $p$-adic Banach representation of ${\text{GL}}_{2}K$ whose locally analytic vectors have an infinitesimal character has Gelfand-Kirillov dimension $\le [K\text{\hspace{0.17em}⁣}:{\mathbf{Q}}_p]$, where $p>2$ and $K$ is a $p$-adic field. We also prove this for the group of units of the quaternions over $K$ replacing ${\text{GL}}_{2}K$. In the process, we make some observations in the theory of $p$-adic Banach representations that might be of independent interest.