  We study a spatial branching model, where the underlying motion is Brownian
motion and the branching is affected by a random collection of reproduction
blocking sets called "mild" obstacles. We show that the quenched local growth
rate is given by the branching rate in the `free' region . When the underlying
motion is an arbitrary diffusion process, we obtain a dichotomy for the local
growth that is independent of the Poissonian intensity. Finally, and most
importantly, we obtain the asymptotics (in probability) of the quenched (when
$d\le 2$) and the annealed (arbitrary d) global growth rates, and identify
subexponential correction terms.

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