Let $X$ be a smooth complex analytic manifold. Let $Z\subset X$ be a smooth compact analytic submanifold. Let $A$ be a holomorphic vector bundle over $X$. Assume that $$H^i(Z, A|_Z)=0 \mbox{ for any } i>0. $$
Is it true that for any open neighborhood $U$ of $Z$ there exists an open neighborhood $V\subset U$ of $Z$ such that $$H^i(V, A)=0 \mbox{ for any } i>0.$$
A reference would be helpful.
Remark. In my case $Z$ is isomorphic to $\mathbb{C}\mathbb{P}^1$ and $A$ is the trivial line bundle. Moreover the normal bundle of $Z$ is isomorphic to direct sum of several copies of $\mathcal{O}(1)$ (I am not sure if it is relevant).