  We consider branching Brownian motion in which initially there is one
particle at $x$, particles produce a random number of offspring with mean $m+1$
at the time of branching events, and each particle branches at rate $\beta =
1/2m$. Particles independently move according to Brownian motion with drift
$-1$ and are killed at the origin. It is well-known that this process
eventually dies out with positive probability. We condition this process to
survive for an unusually large time $t$ and study the behavior of the process
at small times $s \ll t$ using a spine decomposition. We show, in particular,
that the time when a particle gets furthest from the origin is of the order
$t^{5/6}$.
