Ehresmann's lemma equivalent etale topology In complex algebraic geometry we have Ehresmann's lemma which states that given $f:X \to Y$ a smooth surjective proper map between complex manifold this is locally a fibration in the analytic topology.
I was wondering whether there was a generalization of that in the context of etale topology. Lots of statement which are true analytic locally in the context of $\mathbb{C}$ can also be carried in the etale topology for smooth varieties over any algebraically closed field $k$.
In the (beautiful) notes by Conrad https://www.math.ru.nl/personal/bmoonen/Seminars/EtCohConrad.pdf at page 53, he states what he calls etale Ehresmann's lemma: given $f:X \to S$ smooth and proper, the sheaf $R^if_*F$ are locally free whenever $F$ is a torsion etale sheaf on $X$.
The hypothesis are however very general:assuming $X,S$ to be both smooth varieties over an algebraically closed field, it is true the stronger statement that $f$ is a fibration etale locally?
 A: No. I think you might be misinterpretting what 'locally trivial fibration' means.
Namely, what Ehresmann's theorem implies is the following:

Corollary to Ehresmann's theorem: Let $f:X\to Y$ be a smooth proper surjection of complex manifolds. Then, there is an open cover
$\{U_i\}$ of $Y$ such that $f^{-1}(U_i)$ is diffeomorphic (as a manifold over $U_i$) to
$U_i\times B$ for some smooth manifold $B$.

One cannot replace diffeomorphic by 'biholomorphic' or, if $X$ and $Y$ are algebraic and the $U_i$ is a Zariski cover, 'algebraic'.
Here's a silly counterexample to your literal claim.
Consider $D(\Delta)\subseteq \mathbb{A}^2_k=\mathrm{Spec}(k[a,b])$ where, for simplicity, $k$ is an algebraically closed field of characteristic greater than $3$ where
$$\Delta(a,b)=-16(4a^3+27b^2)$$
We then have the universal family of Weierstrass equations living over $D(\Delta)$
$$\mathcal{W}^\text{univ}=V(y^3-axz^2-bz^3)\subseteq \mathbb{P}^2_k\times D(\Delta)$$
We then note that
$$\mathcal{W}^\text{univ}\to D(\Delta)$$
is a family of elliptic curves and so, in particular, smooth proper and surjective. Moreover, each $\mathcal{W}^\text{univ}$ and $D(\Delta)$ is smooth.
That said it's not an etale local fibration since this, in particular, would imply that the fibers of $\mathcal{W}^\text{univ}\to D(\Delta)$ are Zariski locally isomorphic. Indeed, let $\{U_i\to D(\Delta)\}$ be an etale cover such that $\mathcal{W}^\text{univ}\times_{D(\Delta)}U_i\cong U_i\times B_i$ for some $k$-space $B_i$. Then, in particular we see that for all $x\in V_i:=\text{im}(U_i\to D(\Delta))$ (this is an open set) we have that
$$\mathcal{W}^\text{univ}\times_{D(\Delta)}U_i\times_{V_i}x\cong \bigsqcup_{y\mapsto x}B_i$$
which since $k(x)\to k(y)$ is an isomorphism implies that $\mathcal{W}^\text{univ}_x\cong B_i$. Thus, $\mathcal{W}^\text{univ}_x\cong B_i$ for all $x\in V_i$.
Note that we have an algebraic map $j:D(\Delta)\to \mathbb{A}^1_k$, the $j$-function map, which sends, in particular, $x$ to the $j$-invariant $\mathcal{W}^\text{univ}_x$. Since $\mathcal{W}^\text{univ}_x$ is locally isomorphic this implies that $j$ is locally constant which implies, since $D(\Delta)$ is connected, that $j$ is constant and thus $\mathcal{W}^\text{univ}_x$ is actually constant for $x$ in $D(\Delta)$.
Of course, this is ludicrous since $\mathcal{W}^\text{univ}_x$ contains isomorphism classes of every elliptic curve over $k$.
Of course, it is true that the fibration $W^\text{univ}\to D(\Delta)$, when $k=\mathbb{C}$, is analytically locally on the target DIFFEOMORPHIC to the trivial fiber bundle. The operative point being that any two elliptic curves are diffeomorphic but are rarely isomorphic/biholomorphic.
