Picard groups of Fano varieties in positive characteristic Let $k$ be an algebraically closed field of characteristic $p \geq 0$. Let $X$ be a smooth Fano variety over $k$ and let $\ell \neq p$ be a prime.

Is the natural morphism $\mathrm{Pic}(X) \otimes \mathbb{Z}_\ell \to \mathrm{H}^2(X, \mathbb{Z}_\ell(1))$ an isomorphism?

When $p = 0$, this is an easy consequence of Kodaira vanishing and the exponential sequence, together various comparison theorems for étale cohomology. So my question is really about what happens in positive characteristic. For Fano varieties which lift to characteristic zero this is also probably quite easy.
In fact, a positive answer to the following weaker version might be sufficient for my purposes.

Is $\mathrm{rank}(\mathrm{Pic}(X) \otimes \mathbb{Z}_\ell) = \mathrm{rank}( \mathrm{H}^2(X, \mathbb{Z}_\ell(1)))$?

 A: First of all, that cycle class homomorphism is certainly surjective for all Fano manifolds that lift to characteristic $0$.  Secondly, using the usual Kummer sequence, to prove that the cycle class map is surjective, it suffices to prove that the Brauer group is finite.  I think this should not be too hard to prove via the usual argument: for a general point $x_0$ in $X$, and for a dominant, generically finite morphism $u:Y\times \mathbb{P}^1\to X$ that maps $Y\times\{0\}$ to $x_0$, we should be able to prove that every Severi-Brauer variety over $X$ pulls back to a trivial Severi-Brauer on $Y\times \mathbb{P}^1$.  Thus, using restriction / corestriction for $u$, we should get finiteness of the Brauer group.  Of course there is the issue of "rational chain connectedness" instead of "rational connectedness", but I think the extension of this argument should be okay.
Edit.  I should point out, this argument only works if $\ell$ is "sufficiently large".  It would also work for all $\ell$ prime to the characteristic if we tensor the $\mathbb{Z}_{\ell}$-modules with $\mathbb{Q}_{\ell}$.
Second edit. The argument does work integrally for all $\ell$ prime to the characteristic, as Olivier Benoist points out.  I was using the wrong transition maps in computing the inverse limit.
