first order logic – Does FOL extended with least-fixed points satisfy the Compactness Theorem?


I am aware that first-order logics (FOL) satisfies the compactness theorem. That is, if a FOL theory is insatisfiable, a finite subset of the axioms of such theory is insatisfiable too.

Is it the case that FOL extended with least-fixed point (LFP) satisfies the compactness theorem too?