Let $K$ be a field and $C$ a smooth and projective curve over $K$. Then the kernel $Pic^0(C)$ of the degree map injects into $H^0(K,Pic^0_C)$, where $Pic_C^0$ is the connected component of the Picard variety.

I am wondering if there are examples where this is not an isomorphism for $K$ a global field. I am especially interested if there are elements of order prime to $p$ in the cokernel if $K$ is a global field of characteristic $p$.