I have a strong suspicion that yes, but as I am not a specialist in o-minimal structures, I thought that I might have overlooked some corner case. 


The precise statement is as follows: let $X \subset \mathbb{R}^n$ be a set definable in an o-minimal structure. Then there exists a point $P \in X$ and an open neighbourhood $O \subset X$ of $P$ such that $O$ is a closed analytic subset of a real analytic manifold embedded into some open $U \subset \mathbb{R}^n$



Are definable sets in an o-minimal expansion of the real field locally analytic?
No.


In their paper [Quasianalytic Denjoy-Carleman classes and o-minimality](http://www.ams.org/journals/jams/2003-16-04/S0894-0347-03-00427-2/home.html), *J. AMS*, vol. 16 (4), 2003, p. 751—777, Rolin, Speissegger and Wilkie show that there exists a function on $[-1,1]$ which belongs to an o-minimal structure but is nowhere analytic.


Their theorem 1 (p. 752) asserts that for appropriate choices of a sequence $M=(M\_n)$, the class $\mathcal C\_M$ of $\mathcal C^\infty$ functions $f$ such that $\mathopen|f^{(n)}\mathclose|\leq A^{n+1} M\_n$ (for some $A$) generates an o-minimal structure. The precise condition is that $(M\_n)$ satisfies the two conditions:


1. $\sum\_{n=0}^\infty\frac{ M\_n}{M\_{n+1}}=+\infty$;
2. $(M\_n/n!)^2 \leq (M\_{n-1}/(n-1)!) (M\_{n+1}/(n+1)!)$.


By theorem 2 (2), p. 752, there exists such a sequence $(M\_n)$ and a function $f$ in the corresponding class which is nowhere analytic. This furnishes an o-minimal structure without a real analytic cell-decomposition (Corollary, (2), p. 752).


