  In this paper, we study how gaseous dynamical friction (DF) affects the
motion of fly-by stellar-mass black holes (sBHs) embedded in active galactic
nucleus (AGN) discs. We perform 3-body integrations of the interaction of two
co-planar sBHs in nearby, initially circular orbits around the supermassive
black hole (SMBH). We find that DF can facilitate the formation of
gravitationally bound near-Keplerian binaries in AGN discs, and we delineate
the discrete ranges of impact parameters and AGN disc parameters for which such
captures occur. We also report trends in the bound binaries' eccentricity and
sense of rotation (prograde or retrograde with respect to the background AGN
disc) as a function of the impact parameter of the initial encounter. While
based on an approximate description of gaseous friction, our results suggest
that binary formation in AGN discs should be common and may produce both
prograde and retrograde, as well as both circular and eccentric binaries.
