  We construct and study branching fractional Brownian motion with Hurst
parameter $H\in(1/2,1)$. The construction relies on a generalization of the
discrete approximation of fractional Brownian motion (Hammond and Sheffield,
Probability Theory and Related Fields, 2013) to power law P\'olya urns indexed
by trees. We show that the first order of the speed of branching fractional
Brownian motion with Hurst parameter $H$ is $ct^{H+1/2}$ where $c$ is explicit
and only depends on the Hurst parameter. A notion of "branching property" for
processes with memory emerges naturally from our construction.
