The dynamics of the one-dimensional array of magnetic particles (dots) with the easy-plane anisotropy is investigated. The particles interact with each other via the magnetic dipole interaction and the whole system is governed by the set of Landau-Lifshitz equations. The spatially localized and time-periodic solutions known as discrete breathers (or intrinsic localized modes) are identified. These solutions have no analogue in the continuum limit and consist of the core where the magnetization vectors precess around the hard axis and the tails where the magnetization vectors oscillate around the equilibrium position.