Is this homotopy equivalence a deformation retract? Suppose I have a Serre fibration of smooth manifolds $f:X\to Y$ (one may assume $Y$ is an open ball) with a section $s:Y\to X$ of $f\,,$ and furthermore assume the fibers of $f$ are contractible. This implies that $f$ is a homotopy equivalence — can we take this homotopy equivalence to be a deformtation retract onto $s(Y)\,,$ in such a way that points in a fiber stay in the fiber?
If $f:X\to Y$ were a fiber bundle this would be simple, but is it true more generally?
 A: The answer is affirmative. At least modulo one's ability to smooth homotopies and do Morse theory on the manifolds in which you are interested.
We'll start with some generalities which do not quite meet your assumptions. I'll explain below how far they will take you.

Theorem (Heath): Let $X,X'$ be spaces. Assume that $Y$ is a space and there are maps $j:Y\rightarrow X$, $j':Y\rightarrow X'$ and $f:X\rightarrow Y$, $f':X'\rightarrow B$ such that $f\circ j=id_Y=f'\circ j'$. Now suppose that $g:X\rightarrow X'$ is a map with $f\circ j=j'$ and $f'\circ g=f$. If $j,j'$ are closed cofibrations and $f,f'$ are Hurewicz fibrations, then $g$ is a homotopy equivalence under $B$ and over $B$ whenever it is an ordinary homotopy equivalence. $\blacksquare$

The last statement means that under the given assumptions, whenever $g$ is a homotopy equivalence, a homotopy inverse $g':X'\rightarrow X$ satisfying $g'\circ j'=j$ and $f\circ g'=f$ can be found for which there are homotopies $\alpha_t:g'\circ g\simeq id_X$ and $\beta_t:g\circ g'\simeq id_{X'}$ such that $f\circ \alpha_t=f$, $\alpha_t\circ j=j$, $g'\circ \beta_t=g'$ and $\beta_t\circ j'=j'$ for all $t\in I$.
The theorem itself has been taken from  Philip Heath's paper Homotopy Equivalence of a Cofibre Fibre Composite, Can. J. Math., 29, (1977), 1152-1156, where it appears as Theorem 2.2.
In the above, no special assumptions have been made on the spaces. I'll explain now how you get some of the assumptions for free under your conditions. Firstly let's assume that $f:X\rightarrow Y$ is a map between manifolds (not necessary smooth or closed) which is both a Hurewicz fibration and a homotopy equivalence. Applying the homotopy lifting property in the standard way we obtain a map $s:Y\rightarrow X$ which is a homotopy inverse to $f$ satisfying $f\circ s=id_Y$. This $s$ is a topological embedding with closed image (since it has a left inverse and $X$ is Hausdorff).

Proposition: If $X,Y$ are manifolds and $Y$ is embedded as a closed subspace of $X$, then the inclusion $Y\subseteq X$ is a cofibration. $\blacksquare$

We do not need to assume that the embedding has any particular structure other than it is a closed topological embedding. The manifolds may be open or with boundary, and need not even be smooth. The proof is found in Proposition  A.6.7 in Fritsch and Piccinini's book Cellular Structures in Topology.
So now we can apply Heath's Theorem with $X'=Y$. The map $j=s$ is a closed cofibration, as is $j'=id_Y$. The map $f$ is a Hurewicz fibration, as is $f'=id_Y$. Taking $g=f$ Heath supplies us with a homotopy $\alpha_t:s\circ f\simeq id_Y$ such that $f\circ\alpha_t=f$ and $\alpha_t\circ s=s$ for all $t\in I$ (we necessarily have $g'=s$ and that $\beta_t$ is the trivial trivial).
This $\alpha_t$ is the deformation you are looking for.
Now, this solves the problem in the topological category when we assume that $f$ is a Hurewicz fibration. If this condition is acceptable for you, then you can upgrade everything to the smooth category as long as you can smooth the homotopies I have supplied. There is no problem in making both the section $s$ and the homotopy $\alpha_t$ smooth in case $X,Y$ are boundaryless (quote Whitney's Approximation Theorem Th.6.26 in Lee's Introduction to Smooth Manifolds and track through the details in Heath's paper). However I will decline to comment on more general cases.
The final thing to address is how much of the theorem remains true when $f,f'$ are only Serre fibrations. The answer is that everything holds when we impose additionally that $(a)$ $X,X'$ and $Y$ are CW complexes, and $(b)$ $j:Y\subset X$ and $j':Y\subseteq X'$ are subcomplex inclusions. In fact, if $Y$ is metrisable, then assumption $(b)$ not necessary (since $f$ would necessarily be a Hurewicz fibration).
If we are working with smooth closed manifolds, then verifying these assumptions is an easy exercise in Morse theory. Thus, in this case, you can rework all of the above assuming only that $f$ is a Serre fibration. Pinning down the required details in case $X,Y$ are open manifolds or have boundary is again somewhat trickier, and again I will decline to comment.
