Monochromatic infinite sumsets

We show that there is a rational vector space $V$ such that, whenever $V$ is finitely coloured, there is an infinite set $X$ whose sumset $X+X$ is monochromatic. Our example is the rational vector space of dimension $\sup\{\aleph_0,2^{\aleph_0},2^{2^{\aleph_0}},\ldots\,\}$. This complements a result of Hindman, Leader and Strauss, who showed that the result does not hold for dimension below $\aleph_\omega$. So our result is best possible under GCH.