Let $f\colon X\to \mathbb P^1$ be a proper morphism of smooth complex algebraic varieties and let $p\in\mathbb P^1$. Are there a complex disk $\Delta\subseteq\mathbb P^1$ and a Zariski open subset $U\subseteq \mathbb P^1$, with $p\in\Delta\subseteq U$, such that $H^1(f^{-1}(U),{{\mathcal O}^{\rm an}}^\*)\to
H^1(f^{-1}(\Delta),{{\mathcal O}^{\rm an}}^\*)$ is an injection?



Analytic vs Zariski neighbourhood of a fibre
I don't think the modified question works either. Let $E$ be a general elliptic curve. Take $X$ to be the quotient of $E\times \mathbb{P}^{1}$ by an involution which is a translation by a point of order 2 on $E$ and is $z \to 1/z$ on $\mathbb{P}^{1}$. Take the map $f : X \to \mathbb{P}^{1}$ that corresponds to the projection $E \times \mathbb{P}^{1} \to \mathbb{P}^{1}$. Let $p \in \mathbb{P}^{1}$ be a point over which $f$ is smooth, then for any disk $\Delta$ centered at $p$ the analytic Picard group of $f^{-1}(\Delta)$ is just $\text{Pic}(E)$. On the other hand for any Zariski open set $p \in U \subset \mathbb{P}^{1}$ the analytic Picard group of $f^{-1}(U)$ is $\text{Pic}(E')$ where $E'$ is the quotient of $E$ by the point of order 2. Under these identifications the restriction map from the analytic Picard group of $f^{-1}(U)$ to the analytic Picard group of $f^{-1}(\Delta)$ becomes the pullback map  

$\text{Pic}(E') \to \text{Pic}(E)$ which has a kernel $\mathbb{Z}/2$.


