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