Black arsenic phosphorus (b-AsP), as one kind of novel two-dimensional (2D) materials, bridges the band gap between black phosphorus and graphene. Thanks to its great advantages, including high carrier mobility, excellent in-plane anisotropy, and broad tunability band gap, b-AsP has aroused great interest in fields of photonics and photoelectronics. In this paper, ultrathin 2D b-AsP nanomaterials were fabricated by the liquid-phase exfoliation method, and their strong broadband linear and nonlinear absorptions were characterized by ultraviolet-visible-infrared and Z-scan technology. The experimental determination of the nonlinear absorption coefficient and low saturation intensity of b-AsP were -0.23 cm/GW and 3.336 GW/cm2, respectively. Based on density functional theory, the partial charge density and band structure at the conduction band minimum and valence band maximum were calculated, which further proves the excellent optical properties of 2D b-AsP. By first using 2D b-AsP as a novel saturable absorber in both erbium-doped and thulium-doped fiber lasers, mode-locked soliton pulses can stably operate at 1.5 and 2 μm. The laser pulses generated by 2D b-AsP possess higher stability to resist self-splitting than those generated by other 2D material-based mode-lockers. These experimental results highlight that 2D b-AsP has great application potential as a novel optical material in ultrafast photonics from near- to mid-infrared regimes.