It is known that each complex structures on $\mathbb{R}^2$ is biholomorphic to either $\mathbb{C}$ or the open unit disk $\Delta$.

One can continuously deform one complex structure to the other as is for example done in Winkelmann - Deformations of Riemann surfaces (page 3).

My question is

Can this deformation be taken to be holomorphic on the deformation parameter? That is, does there exist a non-trivial complex analytic family $M \to D$ where $D \subset \mathbb{C}$ is a small disk, the central fiber is biholomorphic to $\mathbb{C}$, and the generic fiber is biholomorphic to $\Delta$?

Note that all the theorems that assure complex analytic triviality of deformations when $H^1(X,TX)$ vanishes, use the hypothesis that $X$ is compact.