Let's say I have a surjective, continuous map $f: X \to Y$, and there is a deformation retract of the fibers $f^{-1}(y)$ to a point for every $y \in Y$. Is it always the case that there is a deformation retract of $X$ to a subspace homeomorphic to $Y$? I am looking for relatively easy examples, like $Y = \mathbb{R}^n$ or $Y= S^n$, and $X$ is a subset of $\mathbb{R}^m$ for some $m$.
I guess another way to phrase the wanted property is, does there always exist a section $s:Y \to X$ such that there is a deformation retract of $f^{-1}(y)$ to $s(y)$?
This answer refers to the last part of the question, namely, whether there exists a section with a certain desired property. The answer is no, because a section might not even exist. Let $Y = [0,1]$ be the unit segment and let $X \subseteq [0,1] \times [0,1]$ be the subset $$ X = \left([0,1/2] \times \{0\}\right) \bigcup \left(\{1/2\} \times [0,1]\right) \bigcup \left([1/2,1] \times \{1\}\right) $$ Let $f: X \longrightarrow Y$ be the projection on the first coordinate. Then every fiber of $f$ is contractible but $f$ does not admit a section.
Edit: Here is a another example where $X$ and $Y$ are not even homotopy equivalent. Take $X = [0,1)$, $Y = S^1 \subseteq \mathbb{C}^*$ and $f: X \longrightarrow Y$ the map $f(x) = e^{2\pi i x}$.