I feel like I already answered this question, but it might have been a variant with fibers isomorphic to tori. Let the base $B$ be $\mathbb{A}^2$ with coordinates $s$ and $t$. Begin with $B\times \mathbb{P}^3$, where homogeneous coordinates on $\mathbb{P}^3$ are $[x,y,z,w]$. Let $S$ be the Cartier divisor in $B\times \mathbb{P}^3$ with defining equation $yz-(sx+tw)^2=0$. Let $L$ be the Cartier divisor in $S$ with defining equation $y+z-2(sx+tw)=0$. Let $U$ be the complement of $L$ in $S$. Then $U$ is affine, the morphism $U\to B$ is smooth, and the fiber over every point other than $(0,0)$ is isomorphic to $\mathbb{A}^2$. Of course the fiber over $(0,0)$ is isomorphic to a disjoint union of two copies of $\mathbb{A}^2$. Thus, define $V\subset U$ to be the open subscheme obtained by removing one of these two copies of $\mathbb{A}^2$, i.e., remove the closed subscheme with defining equations $s=t=z=0$. Then $V$ is quasi-affine, and the affine hull is $U$; this follows by Hartog's theorem / the Riemann extension theorem / S2 extension. Therefore $V$ is not isomorphic to an affine space. However, the projection $V\to B$ has all of the requisite properties.

**Edit.** The older answer I mention above was similar, but a little bit different. That answer was in response to the following similar question, When is a holomorphic submersion with isomorphic fibers locally trivial?.