I'm in the situation to have a smooth proper curve $X$ over $Spec(\mathbb C)$, from which I consider the analytification $X^{an}$, which I consider as a compact Riemann surface.

Furthermore I have given a vector bundle $F$ on $X$ with analytification $F^{an}$.

Let $p$ denote a closed point of the curve.

Now I am given an analytic section of $F^{an}$ on the complement of the point $s \in H^0(X^{an}-p, F^{an})$. The question is if there is an algebraic section $\in H^0(X-p,F)$, which goes to $s$ under analytification.