Let $X$ be an Hausdorff space, and let $p,q$ two distinct points. Then if $X$ retracts on $\{ p,q\}$, establish if $X$ has at least two arcwise connected components, or if $X$ has at most two arcwise connected component.
My attemp: A retraction is a continuous surjective map $$r:X\rightarrow \{p ,q\}$$ such that $r\circ i =\operatorname{id}_{ \{ p,q\}}$. Since $\{ p,q\}$ has the discrete topology, we know that, by continuity and surjectivity, $X$ has at least two connected components. On the other hand, we can't have three or more arcwise connected components, because the continuous image of a continous set is connected, hence there are exactly two connected components. What I don't see is why these connected components are indeed arcwise connected, maybe because I am not using the fact that $X$ is Hausdorff nor the fact that $i\circ r$ is omotopic to $\operatorname{id}_{X}$.
This gets it backwards - the pre-images of a connected set need not be connected, so we cannot conclude that the pre-images of $p$ and $q$ are connected.
As noted in the comments by freakish, the fact that a continuous image of a connected set is connected only tells you that $X$ is not connected, and therefore it has at least two connected components.
Next, as you have correctly observed, these components need not be arcwise connected, but that is no matter - since every arcwise component is connected, it is entirely contained in some connected component, and so there are at least two arcwise components. There could of course be more than two, as the example of any discrete space shows.
Finally, the Hausdorff assumption is necessary because to even talk about arcwise components, we need for arc-connectedness to be an equivalence relation, and in general, it need not be (for example, in the line with a doubled origin, $0_1$ is arcwise connected to $1$, which in turn is arcwise connected to $0_2$, but $0_1$ and $0_2$ are not arcwise connected to each other). In the Hausdorff case, this issue is not a concern, since arcwise connectedness is the same as path connectedness, and the latter is always an equivalence relation.
Remark.
Regarding the retraction being homotopic to the identity, that is a stronger condition than simply being a retraction. If the retraction were homotopic to the identity (a so-called “deformation retraction”), then you would then have precisely two arcwise components, since the homotopy would provide a path from each point to either $p$ or $q$.