Suppose we are given a fiber bundle $p:E \to B$ and a point $x \in B$. Denote by $p \big|_{p^{-1}(x)}:p^{-1}(x) \to B$ the restricted fiber bundle and by $\Gamma^0(p)$ (resp. $\Gamma^0(p \big|_{p^{-1}(x)})$) the space of continuous sections with the compact-open topology. Is the restriction map $\rho:\Gamma^0(p)\to\Gamma^0(p\big|_{p^{-1}(x)})$ in general a weak homotopy equivalence or not? I'm obviously having trouble with some basic things but any help is well appreciated.
UPDATE: Just to make clear what I have so far: since we're considering the restriction to one point we have that $\Gamma^0(p\big|_{p^{-1}(x)})\cong p^{-1}(x)$ and thus it seems to me highly unlikely that $\rho$ might be a weak homotopy equivalence in general. Except if I'm missing some very basic fact about fiber bundles which is what I'm here to check for.
UPDATE 2: I just tried to prove surjectivity of the induced map $\rho_*$ on homotopy in a general setting but was unsuccessful, hinting at either a wrong assumption or an inept mathematician.