Which are nontrivial examples of analytical functions on Frechet spaces?

Question

Let $X$ be a real linear topological space, which is a (separable) Frechet space, such that the topology on $X$ is generated by the countable family $\{p_n:n\in\omega\}$ of norms . A real-valued function $f$ is $X$ called analytic, provided $f$ is continuous and the restriction of $f$ on each line of $X$ is an analytic function. We are interested in examples of analytic functions $f$ on $X$ having no $n\in\omega $ such that $f$ is a continuous function on $(X, p_n)$.