Example. $\pi: \{(z,w)\in {\mathbb C}^2: |zw|<1\}\to {\mathbb C}$, $\pi(z,w)=z$.

Example. $\pi: \{(z,w)\in {\mathbb C}^2: |zw|<1\}\to {\mathbb C}$, $\pi(z,w)=z$.

Edit. Similarly, to get a nontrivial deformation of ${\mathbb C}^n$, consider $$ X=\{(z_0, z_1,...,z_n)\in {\mathbb C}^{n+1}: |z_0 z_1|<1\} $$ and let $\pi$ be the projection of $X$ to ${\mathbb C}$ which the 1-st coordinate line in ${\mathbb C}^{n+1}$. Then $\pi^{-1}(t)$ (for $t\ne 0$) will be biholomorphic to $B \times {\mathbb C}^{n-1}$, where $B\subset {\mathbb C}$ is an open disk. Hence, these fibers are not biholomorphic to $\pi^{-1}(0)={\mathbb C}^{n}$.

A better question, I think, is if every Stein manifold of positive dimension admits a nontrivial deformation. I suspect that this is known.