Suppose that for a contractible space $A$ we are given a fiber bundle $p:E\to A$ and denote for $B\subset A$ by $E(B)$ the restricted bundle. I have good reason to believe that in this situation the restriction map $r:\Gamma^0(E)\to\Gamma^0(E(B))$ is a Serre fibration when the section spaces are endowed with the compact-open topology. After a few unsuccessful tries of proving this I decided it might not be a bad idea to ask the denizens of StackExchange for some help. Any hints/ideas/comments are very appreciated.
EDIT: To give a concrete example that bothers me consider $A=D^k\times D^{m-k}$ and $B=D^k_{\frac{1}{2}}\times D^{m-k}$ where $D^k$ denotes the unit disc in $k$ dimensions and $D^k_{\alpha}:=\{x\in D^k\,\big|\,\alpha\leq\|x\|\leq 1\}$.