Let $X$ and $Y$ be *smooth* varieties over the field of complex numbers $\bf C$
(smooth integral separated schemes of finite type over $\bf C$). Let
$$f\colon X\to Y$$
be a surjective morphism such that
for any closed point $y\in Y$, the schematic fibre $f^{-1}(y)\subset X$
is isomorphic to the affine space ${\Bbb A}_{\bf C}^{n(y)}$.
Moreover, assume that the morphism $f$ is *smooth* (which is equivalent to the assumption that $n(y)$
is the constant function $n(y)=n$, where $n=\dim X-\dim Y$).

Consider the real $C^\infty$-manifolds $X^\infty=X({\bf C})$ and $Y^\infty=Y({\bf C})$ and the induced $C^\infty$-map
$$f^\infty\colon X^\infty\to Y^\infty.$$
Since $f$ is smooth, the map $f^\infty$ is a submersion, that is, for any $x\in X^\infty$, the differential
$$d_x f\colon T_x(X)\to T_{f(x)}Y$$
is surjective. Moreover, each fibre of $f^\infty$ is diffeomorphic to ${\bf R}^{2n}$.
By Corollary 31 of G. Meigniez, Submersions, fibrations and bundles,
Trans. Amer. Math. Soc. 354 (2002), no. 9, 3771-3787,
the map $f^\infty$ is a *locally trivial fibre bundle of $C^\infty$-manifolds*, that is, for any $y\in Y^\infty$
there exists an open neighborhood ${\mathcal U}_y$ of $y$ in $Y^\infty$
such that $f^{-1}({\mathcal U}_y)\simeq {\bf R}^{2n}\times {\mathcal U}_y$,
where $\simeq$ denotes a $C^\infty$-diffeomorphism compatible with the projections onto ${\mathcal U}_y$.

Question 1.Does it follow that the morphism $f$ is a locally trivial fibre bundle in the étale topology, that is, for any closed point $y\in Y$ there exists an étale open neighborhood $ U_y\to Y$ of $y$ such that $$X\times_Y U_y\simeq {\Bbb A}_{\bf C}^n\times_{\bf C} U_y\,,$$ where $\simeq$ denotes an isomorphism of $\bf C$-varieties compatible with the projections onto $U_y$ ?

Question 2.Is $f$ a locally trivial fibre bundle in the flat topology?

Consider the complex analytic manifolds $X^{\rm an}=X({\bf C})$, $Y^{\rm an}=Y({\bf C})$ and the induced complex analytic morphism $$f^{\rm an}\colon X^{\rm an}\to Y^{\rm an}.$$

Question 3.Is $f^{\rm an}\colon X^{\rm an}\to Y^{\rm an}$ a locally trivial fibre bundle of complex analytic manifolds, that is, for any $y\in Y^{\rm an}$ there exists an open neighborhood ${\mathcal U}_y$ of $y$ in $Y^{\rm an}$ such that $(f^{\rm an})^{-1}({\mathcal U}_y)\simeq {\bf C}^n\times {\mathcal U}_y$, where $\simeq$ denotes an analytic isomorphism compatible with the projections onto ${\mathcal U}_y$ ?