Pull-back of line bundles and field extension

Question

Let $X$ be a smooth, projective variety over a field $K$ of characteristic $0$ (not necessarily algebraically closed) and $L$ an invertible sheaf on $X_{\bar{K}}=X \times_K \mbox{Spec}(\bar{K})$, where $\bar{K}$ is the algebraic closure of $K$. Does there exist a finite field extension $K'$ of $K$ and an invertible sheaf $L'$ on $X_{K'}:=X \times_K \mbox{Spec}(K')$ such that $L' \otimes_{K'} \bar{K} \cong L$ (in other words, the base change of $L'$ to $X_{\bar{K}}$ is $L$)? If so, is this true if $K$ is a perfect field of characteristic $p>0$. Any reference/hint is most welcome.