Is there an equivalent to the DLP with extension fields?

Question

For instance, if I have an extension field of $p^n$, is there a way to recover $p$, other than brute force checking?

Answer 1

Of course. There is also a way to recover $p$ in fields of prime order other than brute force. All of these algorithms are faster than brute force. Yet, they are still not practical for real-world attacks.

So, if you meant for real world attacks, then then answer is yes, if the field has small characteristic. See for example this paper and this list of papers.

For large characteristic fields, the answer is no (otherwise elliptic curve cryptography would be dead).