Assume the following condition for the ring T = F_p[[X,S]]/I:
Condition 1. T is NOT a zero ring. Condition 2. I is generated by 3 elements of F_p[[X,S]], but NOT by 2 elements.
Then, is T a gorenstein local ring?
No. Take $I=\langle X^2,XS, S^2\rangle$. Then $T$ is graded, $T=T_0\oplus T_1$, with $\dim_{\,\Bbb{F}_p}T_1=2$. But for a graded artinian Gorenstein ring $T_0\oplus \ldots \oplus T_d$ one has $\dim T_d=1$ (and the pairing $T_p\times T_{d-p}\rightarrow T_d$ is perfect).
In addition to @abx's answer, these rings are not Gorenstein.
E. Kunz, Almost complete intersections are not Gorenstein, J. Alg. 28 (1974), 111–-115.