Is a zero mean curvature submanifold, with a flat open subset, flat everywhere?

Question

Let $\Sigma \subset \mathbb{R}^{n+1}$ be a complete and connected minimal submanifold (with or without boundary), i.e. its mean curvature $H$ vanishes everywhere. Assume also that there exists an open subset $A \subset \Sigma$ contained in a hyperplane, i.e. where the second fundamental form vanishes.

Can we say that the second fundamental form vanishes everywhere in $\Sigma$ ?

Any help would be very appreciated.

Answer 1

Lemma 1 here: Minimal submanifolds are real analytic.