Can $y^2=x^3-x+1$ elliptic curve with $GF(3^m)$ where $m=97$ be used for Diffie Hellman key exchange?
This curve is thoroughly insecure. These researchers performed a computation to break discrete log on this exact curve.
All small characteristic pairing-friendly curves are insecure under modern knowledge. Here is another paper breaking discrete log on a curve over $\operatorname{GF}(3^{6\cdot 509})$ -- note that this field size is much bigger than your curve.