Is the analytic version of the Whitney Approximation Theorem true?
The result is not stated in Grauert's paper. On the other hand, Grauert proves that every real analytic manifold $M$ sits as a real analytic totally real submanifold, and analytic deformation retraction, in a Stein manifold $M_{\mathbb{C}}$. So every continuous map $\phi \colon M \to N$ of real analytic manifolds extends to a continuous map $\phi \colon M_{\mathbb{C}} \to N_{\mathbb{C}}$. Since $M_{\mathbb{C}}$ and $N_{\mathbb{C}}$ are Stein manifolds, Oka's theorem proves that this continuous map is homotopic to a holomorphic map. This then composes with the real analytic inclusion $M \to M_{\mathbb{C}}$ and real analytic deformation retraction $N_{\mathbb{C}} \to N$, so $\phi$ is homotopic to a real analytic map. For me, at least, this is not obviously contained in Grauert's paper (or at least I didn't spot it), although I am sure that Grauert would have seen it as a consequence.
According to this paper by Michael Langenbruch, this was proved by none other than H. Whitney. The paper has lots of references.