Is there a function which is analytic at all points of $\mathbb{C}^{n}$ and maps $\mathbb{R}^{n}$ into $\mathbb{R}$ but its restriction to $\mathbb{R}^{n}$ has no real power series representation around a real point?
Similar question when the function in question is complex analytic in a neighborhood of the origin and it maps the real points of that neighborhood into the reals but its restriction to the real points has no real power series representation in a neighborhood of the origin.
Obviously a positive answer to the first question implies a positive answer to the second question.
Power series are Taylor series. Existence of complex derivatives implies existence or real derivatives (limit in a subset). Applying the hypothesis, the values of derivatives are real, .i.e., the coefficients of the series are real and you have real power series representation.