  We prove a global well-posedness result for the Landau-Lifshitz equation with
Gilbert damping provided that the BMO semi-norm of the initial data is small.
As a consequence, we deduce the existence of self-similar solutions in any
dimension. In the one-dimensional case, we characterize the self-similar
solutions associated with an initial data given by some ($\mathbb{S}^2$-valued)
step function and establish their stability. We also show the existence of
multiple solutions if the damping is strong enough. Our arguments rely on the
study of a dissipative quasilinear Schr\"odinger obtained via the stereographic
projection and techniques introduced by Koch and Tataru.
