No.

If $\mathbb Z_{p^{\infty}}$ is a Prufer group for prime $p$, then its endomorphism ring is isomorphic to the ring $\mathbb Z_p$ of $p$-adic integers. Hence $\mathbb Z_{p^{\infty}}$ is a $\mathbb Z_p$-module, and we can form the idealization $R:=\mathbb Z_p\oplus \mathbb Z_{p^{\infty}}$. The ideals of this ring are known: they are of the form $0\oplus H$ for a subgroup $H\leq \mathbb Z_{p^{\infty}}$ or of the form $(p^k)\oplus \mathbb Z_{p^{\infty}}$ for some nonnegative integer $k$.
(See Example 1 of Paul A. Froeschl, Chained rings. Pacific J. Math.
Volume 65, Number 1 (1976), 47-53.)
In particular, $R$ is a local ring with exactly one non-finitely generated ideal $I:=0\oplus \mathbb Z_{p^{\infty}}$. This ideal is prime because $R/I$ is isomorphic to $\mathbb Z_p$, a domain.