Plasmonic materials provide electric-field localization and light confinement at subwavelength scales due to strong light-matter interaction around resonance frequencies. Graphene has been recently studied as an atomically thin plasmonic material for infrared and terahertz wavelengths. Here, we theoretically investigate localized surface plasmon resonances (LSPR) in a monolayer, nanostructured black phosphorus (BP). Using finite-difference time-domain simulations, we demonstrate LSPRs at mid-infrared and far-infrared wavelength regime in BP nanoribbon and nanopatch arrays. Because of strong anisotropic in-plane properties of black phosphorus emerging from its puckered crystal structure, black phosphorus nanostructures provide polarization dependent, anisotropic plasmonic response. Electromagnetic simulations reveal that monolayer black phosphorus nanostructures can strongly confine infrared radiation in an atomically thin material. Black phosphorus can find use as a highly anisotropic plasmonic devices.