Let $\xi = \pi \colon E \to B$ a topological fiber bundle with connected base $B$, $E_x = \pi^{-1}(x)$ the fiber at $x \in B$, $j \colon E_x \hookrightarrow E$ the canonical injection, and let suppose that there exists a retraction $r \colon E \to E_x$, i.e. $r◦j=Id_{E_x}$. Can we conclude that $\xi$ is trivial ?

I found this proposition somewhere in a book or a pdf, without proof, but can't remember where. I thought I had a proof but it is plain wrong. The converse is indeed true (i.e. if $\xi$ is trivial, $E \cong B \times F$, there is a retraction of $E$ on any fiber), and this proposition seems intuitively correct... but is it ?