I am researching different generalizations of the familiar open mapping theorem from functional analysis. Every "proof" I attempt while simply assuming positive-homogeneity, even in the finite-dim case, has hit a giant brick wall, making me think it is not true, but I cannot seem to write down a counter-example.

Is there an example in which a map $f\colon\mathbb{R}^m\longrightarrow\mathbb{R}^n$ is positively-homogeneous, onto, and not open?

For simplicity, why not take $f$ to be analytic and homogeneous of degree $k\in\mathbb{N}$ and $m\geq n$?