Let me construct a non-paracompact counterexample. Of course, a paracompact counterexample is better in many respects since differential topology is usually restricted to paracompact manifolds and because the manifolds that naturally occur and the ones sold here are all paracompact.

Let $L=([0,1)\times\omega_{1})\setminus\{(0,0)\}$ be the long line. Let $M$ be any paracompact manifold with the fixed point property for homeomorphisms. For instance, $M$ could be a projective space over the reals, the complex numbers, or over the quaternions.
Let $N=L\times M$.

I claim that $N$ has the fixed point property for homeomorphisms but not for continuous maps.

Clearly, there is a retraction $f:N\rightarrow S$ where $S\simeq M\times(0,1]$. The set $M\times(0,1]$ does not have the fixed point property since the map $M\times(0,1]\rightarrow M\times(0,1],(x,y)\mapsto(x,y/2)$ has no fixed point. Therefore there is some map $g:S\rightarrow S$ without a fixed point. Thus $g\circ f:N\rightarrow S$ has no fixed point.

Now assume that $h:N\rightarrow N$ is a homeomorphism.

If $x\in M$, then let $\Gamma_{x}:L\rightarrow L,\Delta_{x}:L\rightarrow L$ be the maps where $\Gamma_{x}(y)=\pi_{1}(h(y,x)),\Delta_{x}(y)=\pi_{1}(h(y,x))$. Let
$C_{x}=\{\alpha\in\omega_{1}|\Gamma_{x}(\alpha,0)=(\alpha,0)\},D_{x}=
\{\alpha\in\omega_{1}|\Delta_{x}(\alpha,0)=(\alpha,0)\}.$

I claim that the sets $C_{x},D_{x}$ are club sets (club stands for closed unbounded). It is clear that the sets $C_{x},D_{x}$ are closed, so we now need to show $C_{x},D_{x}$ are unbounded.

It is easy to see that for all $\alpha\in\omega_{1}$, we have $\Gamma_{x}(z)>(0,\alpha)$ for sufficiently large $z$. Therefore, for all $\alpha\in\omega_{1}$ there is a sequence $(y_{n})_{n}$ with $(0,\alpha)<y_{0}$ and where $y_{n}<y_{n+1},y_{n}<\Gamma_{x}(y_{n+1}),\Gamma_{x}(y_{n})<y_{n+1},\Gamma_{x}(y_{n})<y_{n+1}$. Furthermore, one can choose the sequence $(y_{n})_{n}$ so that $\lim_{n\rightarrow\infty}y_{n}=(0,\gamma)$ for some $\gamma$. Therefore, we have $\Gamma_{x}(0,\gamma)=\lim_{n\rightarrow\infty}\Gamma_{x}(y_{n})=(0,\gamma)$. We conclude that $\gamma\in C_{x}$, so $C_{x}$ is unbounded. Thus $C_{x}$ is a club set. The set $D_{x}$ is a club set as well by an identical argument.

Let $A\subseteq M$ be a countable dense subset. Then let $E=\bigcap_{x\in X}C_{x}\cap D_{x}$. Then $E$ is a club set being the countable intersection of club sets.

Now assume that $\alpha\in E$. Then since $(\alpha,0)=\Delta_{x}(\alpha,0)=\pi_{1}(h((\alpha,0),x)$ for $x\in X$, we have
$h[\{(\alpha,0)\}\times X]\subseteq\{(\alpha,0)\}\times M$. Therefore, we have
$h[\{(\alpha,0)\}\times M]=h[\overline{\{(\alpha,0)\}\times X}]\subseteq\{(\alpha,0)\}\times M$. Similarly, we have $h^{-1}[\{(\alpha,0)\}\times M]\subseteq\{(\alpha,0)\}\times M$. Therefore, $\{(\alpha,0)\}\times M\subseteq h[\{(\alpha,0)\}\times M]$. We conclude that
$h[\{(\alpha,0)\}\times M]=\{(\alpha,0)\}\times M$.

However, since $M$ has the fixed point property for homeomorphisms, and $h[\{(\alpha,0)\}\times M]=\{(\alpha,0)\}\times M$, the mapping $h$ has some fixed point on $\{(\alpha,0)\}\times M$.

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