  The branching Brownian sausage in $\mathbb{R}^d$ was defined by Engl\"ander
in [Stoch. Proc. Appl. 88 (2000)] similarly to the classical Wiener sausage, as
the random subset of $\mathbb{R}^d$ scooped out by moving balls of fixed radius
with centers following the trajectories of the particles of a branching
Brownian motion (BBM). We consider a $d$-dimensional dyadic BBM, and study the
large-time asymptotic behavior of the volume of the associated exponentially
shrinking branching Brownian sausage (BBM-sausage). Using a previous result on
the density of the support of BBM, and some well-known results on the classical
Wiener sausage and Brownian hitting probabilities, we obtain almost sure limit
theorems as time tends to infinity on the volume of the shrinking BBM-sausage
in all dimensions.
