  We discover a new class of topological solitons. These solitons can exist in
a space of infinite volume like, e.g., $\mathbb{R}^n$, but they cannot be
placed in any finite volume, because the resulting formal solutions have
infinite energy. These objects are, therefore, interpreted as totally
incompressible solitons.
  As a first, particular example we consider (1+1) dimensional kinks in
theories with a nonstandard kinetic term or, equivalently, in models with the
so-called runaway (or vacummless) potentials. But incompressible solitons exist
also in higher dimensions. As specific examples in (3+1) dimensions we study
Skyrmions in the dielectric extensions both of the minimal and the BPS Skyrme
models. In the the latter case, the skyrmionic matter describes a completely
incompressible topological perfect fluid.
