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Let $\mathcal{N}$ be a non-Archimedean ordered field extension of the realnumbers that is real closed and Cauchy complete in the topology induced by theorder, and whose Hahn group is Archimedean. In this paper, we first review theproperties of weakly locally uniformly differentiable (WLUD) functions, $k$times weakly locally uniformly differentiable (WLUD$^k$) functions, andWLUD$^\infty$ functions at a point or on an open subset of $\mathcal{N}$. Thenwe show under what conditions a WLUD$^\infty$ function at a point$x_0\in\mathcal{N}$ is analytic in an interval around $x_0$, that is, it has aconvergent Taylor series at any point in that interval.  We generalize the concepts of WLUD$^k$ and WLUD$^\infty$ to functions from$\mathcal{N}^n$ to $\mathcal{N}$, for any $n\in\mathbb{N}$. Then we formulateconditions under which a WLUD$^\infty$ function at a point $\boldsymbol{x_0}\in \mathcal{N}^n$ is analytic at that point.

On the analyticity of WLUD$^\infty$ functions of one variable and  WLUD$^\infty$ functions of several variables in a complete non-Archimedean  valued field