  We study $(2,2)$ and $(4,4)$ supersymmetric theories with superspace higher
derivatives in two dimensions. A characteristic feature of these models is that
they have several different vacua, some of which break supersymmetry. Depending
on the vacuum, the equations of motion describe different propagating degrees
of freedom. Various examples are presented which illustrate their generic
properties. As a by-product we see that these new vacua give a dynamical way of
generating non-linear realizations. In particular, our 2D $(4,4)$ example is
the dimensional reduction of a 4D $N=2$ model, and gives a new way for the
spontaneous breaking of extended supersymmetry.
