$I-f$ is a nonlinear homeomorphism on an infinite dimensional Banach space. $P$ is a linear projection. Is $I-Pf$ a homeomorphism?

Question

$f:X\to X$ is a nonlinear operator where $X$ is an infinite dimensional Banach space, $I-f$ is a homeomorphism, and $P:X\to X$ is a linear projection. Additionally, we may assume that $I-Pf$ is proper (I am able to prove this for my specific $f$, but can go no further.)

I would like to know if $I-Pf$ is a homeomorphism, or even a bijection. Is it possible to tell with this information? Thanks!

Edit: I can come up with counterexamples when $I-Pf$ is not proper (e.g. $f=\left(\begin{array}{cc} 1 & -1\\ 1 & 1 \end{array}\right)$ and $P=\left(\begin{array}{cc} 1 & 0\\ 0 & 0 \end{array}\right)$ in the finite dimensional case) so I suppose the question is really whether properness is enough to ensure bijection.