Does $\mathbb R P^n$ retract to $\mathbb R P^k$? I was trying to prove that no such retraction exists. For $k$ odd this follows directly from the (co-)homology groups of these spaces. However, for $k$ even I have no idea how to proceed and if it is even true.

This is true as long as $0k$, then there is no injective homomorphism of graded rings $\mathbb{Z}/2[x]/(x^{k+1})\to \mathbb{Z}/2[x]/(x^{n+1})$ (since $x$ must map to an element $y$ such that $y^{k+1}=0$, and the only such element in degree $1$ is $y=0$).

(I assume you are interested only in the case $k\leq n$, where $\mathbb{R}P^k$ is a subspace of $\mathbb{R}P^n$ in the canonical way. If $k>n$, then $\mathbb{R}P^k$ does not even embed in $\mathbb{R}P^n$.)