  The production of $B$ mesons at forward rapidities is strongly sensitive to
the behavior of the gluon and bottom distribution functions for small and large
values of the Bjorken-$x$ variable. In this exploratory study, we estimate the
cross-section for the $B^{\pm}$ meson production in the kinematic range probed
by the LHCb detector and that will be analyzed by the future Forward Physics
Facility (FPF) considering the hybrid formalism, the solution of the running
coupling Balitsky-Kovchegov equation and distinct description of the bottom
distribution function. We assume an ansatz for the intrinsic bottom component
in the proton wave function, and estimate its impact on the transverse
momentum, rapidity and Feynman-$x$ distributions. Our results indicate that the
presence of an intrinsic bottom strongly modifies the magnitude of the
cross-section at ultra-forward rapidities ($y \ge 6$), which implies an
enhancement of the $B^{\pm}$ production at the FPF. Possible implications on
the prompt neutrino flux at ultra-high energies are also briefly discussed.
