The intrinsic nonlinearities of the spin dynamics in condensed matter systems give rise to a rich phenomenology that can be strongly affected by topology. Here, we study formation of magnonic solitons in the topologically nontrivial band gap of a spin lattice realization of the Haldane model, in both static and dynamic (Floquet) regimes. We consider nonlinearities caused by magnetic crystalline anisotropy and magnon-magnon interactions. We find soliton formation power thresholds as a function of anisotropy coefficient and interaction strength. We predict different classes of topological solitons for the same topological class of the underlying lattice and explain it in terms of a transition from a topologically nontrivial mass to a trivial one. Our findings imply that a soliton can phase separate, containing boundaries between topologically trivial and nontrivial phases, which is associated with a vanishing spin-wave gap.