S.S. Kim and K.H. Kwon gave an explicit example of a monotone smooth but nowhere analytic function (link), which is an anti-derivative of the function $$\psi(x)=\sum\limits_{k=1}^{\infty} \frac{1}{k!}\phi(2^k(x-[x])),$$ where $$\phi(x) = \begin{cases} \exp{\left(-\frac{1}{x^2}-\frac{1}{(x-1)^2}\right)},\qquad & 0 < x < 1,\ \\\ \\\ 0, & \mbox{otherwise.} \end{cases}$$

In fact, the set of smooth but nowhere analytic functions on $\mathbb R$ is of the second category in $C^{\infty}(\mathbb R)$ (just like the set of all continuous but nowhere differentiable functions is of the second category in $C(\mathbb R)$). See a one page note by R. Darst "Most infinitely differentiable functions are nowhere analytic".

S.S. Kim and K.H. Kwon gave an explicit example of a monotone smooth but nowhere analytic function (link), which is an anti-derivative of the function $$\psi(x)=\sum\limits_{k=1}^{\infty} \frac{1}{k!}\phi(2^k(x-[x])),$$ where $$\phi(x) = \begin{cases} \exp{\left(-\frac{1}{x^2}-\frac{1}{(x-1)^2}\right)},\qquad & 0 < x < 1,\ \\\ \\\ 0, & \mbox{otherwise.} \end{cases}$$

In fact, the set of smooth but nowhere analytic functions on $\mathbb R$ is of the second category in $C^{\infty}(\mathbb R)$ (just like the set of all continuous but nowhere differentiable functions is of the second category in $C(\mathbb R)$). See a one page note by R. Darst "Most infinitely differentiable functions are nowhere analytic".

Edit. Kim and Kwon mention in their paper that the first concrete example of smooth
but nowhere analytic function dates back to A Pringsheim ("Zur Theorie der Taylor'schen Reihe und der analytischen Functionen mit beschränktem Existenzbereich." Math. Ann. 42 (1893), no. 2, 153–184.)

S.S. Kim and K.H. Kwon gave an explicit example of a monotone smooth but nowhere analytic function (link), which is an anti-derivative of the function $$\psi(x)=\sum\limits_{k=1}^{\infty} \frac{1}{k!}\phi(2^k(x-[x])),$$ where $$\phi(x) = \begin{cases} \exp{\left(-\frac{1}{x^2}-\frac{1}{(x-1)^2}\right)},\qquad & 0 < x < 1,\ \\\ \\\ 0, & \mbox{otherwise.} \end{cases}$$

In fact, the set of smooth but nowhere analytic functions on $\mathbb R$ is of the second category in $C^{\infty}(\mathbb R)$ (just like the set of all continuous but nowhere differentiable functions is of the second category in $C(\mathbb R)$).

S.S. Kim and K.H. Kwon gave an explicit example of a monotone smooth but nowhere analytic function (link), which is an anti-derivative of the function $$\psi(x)=\sum\limits_{k=1}^{\infty} \frac{1}{k!}\phi(2^k(x-[x])),$$ where $$\phi(x) = \begin{cases} \exp{\left(-\frac{1}{x^2}-\frac{1}{(x-1)^2}\right)},\qquad & 0 < x < 1,\ \\\ \\\ 0, & \mbox{otherwise.} \end{cases}$$

In fact, the set of smooth but nowhere analytic functions on $\mathbb R$ is of the second category in $C^{\infty}(\mathbb R)$ (just like the set of all continuous but nowhere differentiable functions is of the second category in $C(\mathbb R)$). See a one page note by R. Darst "Most infinitely differentiable functions are nowhere analytic".