Given a finitely generated field $F$ with prime field $k$, we assume $k$ is finite, of characteristic $p$. Fix a prime $\ell$ invertible in $k$.

In the discussion right after [K, Lemma 2.3], the variant

$$H^{*,\star}:=\tilde{\rm H}^*_{\rm cont}(\text{Spec}(F)/\text{Spec}(k),\mathbf{Z}_{\ell}(\star))$$ of Jannsen's continuous étale cohomology of $\text{Spec}(F)$ is defined, for integers $(*,\star)$, with $*$ nonnegative (notation as in loc cit, with $X = \text{Spec}(F)$ and $S = \text{Spec}(k)$).

A more explicit description of $H^{*,\star}$ can be given, as per loc cit, as follows.

Write $F$ as a filtered colimit of the global complete intersection $k$-subalgebras $A$ of $F$ (see [SP, Tag 07BV)], and denote by ${\rm H}_{\rm cont}^*(\text{Spec}(A),\mathbf{Z}_{\ell}(\star))$ the continuous étale cohomology of Jannsen's.

Remark. Here $\mathbf{Z}_{\ell}(\star)$ is the pro-sheaf $\{\mu_{\ell^n}^{\otimes\star}, n\ge 1\}$, as per construction of Jannsen's continuous étale cohomology. See [J] for details on continuous étale cohomology, and [K, $\S$2] for an overview. Roughly, continuous étale cohomology of a scheme $X$ with coefficients in the pro-sheaf $\mathbf{Z}_{\ell}(\star)$ is, degree-wise, an extension of usual $\ell$-adic cohomology of $X$ by a $\lim^1$ module.

Then the $H^{*,\star}$ from above can be written as $$H^{*,\star} = \varinjlim_{k\subset A\subset F}{\rm H}_{\rm cont}^*(\text{Spec}(A),\mathbf{Z}_{\ell}(\star)).$$

Now define $$H^{*,\star}_{\mathbf{Q}} := H^{*,\star}\otimes_{\mathbf{Z}}\mathbf{Q}.$$

Question. Why does $H^{i,p}_{\mathbf{Q}}$ vanish for $i>p+1$, regardless of the transcendence degree of $F$?

Example. When $F$ is a finite extension of $k$, then we get $$H^{*,\star} = {\rm H}^*_{\rm cont}(F,\mathbf{Z}_{\ell}(\star)).$$ Here the right side is continuous cohomology of the field $F$ as in [J, Thm. 3.2]. Since $F$, in this example, is finite, we get that, for all integers $p$, $H^{i,p}$ is zero for all $i\neq 1$, and finite for $i=1$, so $H^{i,p}_{\mathbf{Q}}=0$ for all $i$.

References:

[J] U. Jannsen, Continuous étale cohomology. Math. Annalen.
[K] B. Kahn, A sheaf-theoretic reformulation of the Tate conjecture. Preprint.
[SP] Stacks Project.