Are there any (asymmetric) cryptographic primitives not relying on arithmetic over prime fields and/or finite fields?
Braid cryptography?  Knapsack cryptosystems, like Nasako–Murikami?

Lattice-based cryptography tends to work in polynomial rings or modules with coefficients in finite fields, but whose higher-level structure is not a field.

Also: don't forget RSA!  RSA works in a ring, not a field.