Usage of GF(p^m) fields, where p != 2
Yes, cryptosystems like ElGamal or Shnorr based on the intractability
of Dlog Problem are are indicated to be implemented on finite field,
which is not the case of the RSA for which a model was proposed in the
early $80^{ies}$, and immediatly broken. As you know, a finite field is denoted by $GF(q)$ where $q=p^m$ and p would be any Prime. But in the case there is no special advantage to use such q with $p\neq 2$, because the arithmetic on this field is a bit more complicated, and each party would beforehand agree on using the same irreducible polynomial for building the corresponding splitting field.

However, other applications could use these fields for specific attacks, as the MOV attack which was introduced to reduce the Dlog Problem on supersingular curves.