  We consider the limiting extremal process ${\mathcal X}$ of the particles of
the binary branching Brownian motion. We show that after a shift by the
logarithm of the derivative martingale $Z$, the rescaled "density" of
particles, which are at distance $n+x$ from a position close to the tip of
${\mathcal X}$, converges in probability to a multiple of the exponential $e^x$
as $n\to+\infty$. We also show that the fluctuations of the density, after
another scaling and an additional random but explicit shift, converge to a
$1$-stable random variable. Our approach uses analytic techniques and is
motivated by the connection between the properties of the branching Brownian
motion and the Bramson shift of the solutions to the Fisher-KPP equation with
some specific initial conditions initiated in \cite{BD1,BD2} and further
developed in the present paper. The proofs of the limit theorems for ${\mathcal
X}$ rely crucially on the fine asymptotics of the behavior of the Bramson shift
for the Fisher-KPP equation starting with initial conditions of "size"
$0<\varepsilon\ll 1$, up to terms of the order $[{(\log
\varepsilon^{-1})]^{-1-\gamma}}$, with some $\gamma>0$.
