For a compact Riemann surface $B$ of genus $\geq 2$, it is a consequence of the Narasimhan-Seshadri theorem that there exist rank-$2$ vector bundles $E \to B$ of degree zero, all of whose symmetric powers are stable. As Mumford observed, such vector bundles give rise to ruled complex surfaces ($\mathbb{P}^1$-bundles over $B$) $X := \mathbb{P}(E) \to B$ with a line bundle $L := \mathcal{O}_{X/B}(1)$ having $L.C > 0$ on every curve, yet $L^2 = 0$.

Is it known whether such bundles $E$ exist in characteristic $p$ also? If not, is there any example known of a surface $X/k$ in characteristic $p$ with a line bundle $L \to X$ having $L^2 = 0$ yet $L.C > 0$ on every curve $C \subset X$?

For a compact Riemann surface $B$ of genus $\geq 2$, it is a consequence of the Narasimhan-Seshadri theorem that there exist rank-$2$ vector bundles $E \to B$ of degree zero, all of whose symmetric powers are stable. As Mumford observed, such vector bundles give rise to ruled complex surfaces ($\mathbb{P}^1$-bundles over $B$) $X := \mathbb{P}(E) \to B$ with a line bundle $L := \mathcal{O}_{X/B}(1)$ having $L.C > 0$ on every curve, yet $L^2 = 0$.

Is it known whether such bundles $E$ exist in characteristic $p$ also? If not, is there any example known of a surface $X/k$ in characteristic $p$ with a line bundle $L \to X$ having $L^2 = 0$ yet $L.C > 0$ on every curve $C \subset X$?

Update. I would also like to inquire if such a construction has been made in Arakelov geometry -- for arithmetic surfaces, or adelic line bundles on curves, or on higher dimensional arithmetic varieties. It would be enough to have a number field $K$ and a rank-$2$ degree-zero hermitian vector bundle $E$ on $\mathrm{Spec}(O_K)$ having all its symmetric powers stable. The existence of such an $E$ appears to be far from trivial, however, as it is not even known that semistable hermitian bundles are closed under the tensor product (this has been conjectured by Soule).