I initially asked this question on MSE but I haven't had any luck.

The Whitney Approximation Theorem states that any continuous map between smooth manifolds is homotopic to a smooth map. If the manifolds are real analytic, is every continuous map between them homotopic to a real analytic map?

I know that the natural generalisation to complex manifolds fails. That is, not every continuous map between complex manifolds is homotopic to a holomorphic map.