Let $K$ be a field of characteristic $p>0$ and $C$ a regular projective geometrically integral curve over $K$.

If $C$ is smooth, then ${\rm Pic}^0(C)$ is isomorphic to the Jacobian $J_C$, so in particular the $n$-torsion ${\rm Pic}^0(C)[n]$ is finite for every $n$, including $n=p$.

Question: Is ${\rm Pic}^0(C)[p]$ finite also when $C$ is regular but not smooth?

Of course, $K$ is then necessarily imperfect. From what I understand (e.g. from Chapters 8 and 9 of the book by Bosch-Lütkebohmert-Raynaud), in this case ${\rm Pic}^0(C)$ is still a group scheme, possibly with a unipotent part, but not containing a copy of $\mathbb{G}_a$ (in particular non-split). However, that in itself does not seem to be sufficient to conclude finiteness of the $p$-torsion.

Let $K$ be a field of characteristic $p>0$ and $C$ a regular projective geometrically integral curve over $K$.

If $C$ is smooth, then the connected component ${\rm Pic}^0_C$ of the Picard scheme of $C$ is isomorphic to the Jacobian $J_C$, so in particular the $n$-torsion of the class group of $C$, ${\rm Cl}(C)[n]={\rm Pic}^0_C(K)[n]=J_C(K)[n]$, is finite for every $n$, including $n=p$.

Question: Is ${\rm Cl}(C)[p]={\rm Pic}^0_C(K)[p]$ finite also when $C$ is regular but not smooth?

Of course, $K$ is then necessarily imperfect. From what I understand (e.g. from Chapters 8 and 9 of the book by Bosch-Lütkebohmert-Raynaud), in this case ${\rm Pic}^0_C$ is still a group scheme, possibly with a unipotent part, but not containing a copy of $\mathbb{G}_a$ (in particular non-split). However, that in itself does not seem to be sufficient to conclude finiteness of the $p$-torsion.