Is there no norm in $C^\infty ([a,b])$?

Literally, the claim is wrong, since the space has dimension $2^{\aleph_0}$, and there are Banach spaces with the same dimension (e.g. the $\ell^p(\mathbb{N})$ spaces). Since the rationals are dense in $[a,b]$, there is a linear injection of $C^\infty([a,b])$ into $\mathbb{R}^{\mathbb{Q}\cap [a,b]}$, and the latter space has cardinality

$$\operatorname{card}(\mathbb{R})^{\aleph_0} = \left(2^{\aleph_0}\right)^{\aleph_0} = 2^{\aleph_0},$$

so $\dim C^\infty([a,b]) \leqslant 2^{\aleph_0}$. On the other hand, the functions $t\mapsto e^{ct},\, c\in\mathbb{R}$ are linearly independent, so the dimension is at least $2^{\aleph_0}$.

What is meant, even if it is not said, when that claim is made, is that the space $C^\infty([a,b])$ in its natural topology - the usual Fréchet space topology induced by the seminorms $\lVert f\rVert_k = \sup \{ \lvert f^{(k)}(t)\rvert : t\in [a,b]\}$ for $k\in\mathbb{N}$ - is not normable.

An easy way to see that is to note that $C^\infty([a,b])$ is a Fréchet-Montel space, that is, every closed and bounded subset is compact. That is a repeated application of the Ascoli-Arzelà theorem; the boundedness of $\{f^{(k+1)} : f\in B\}$ implies the equicontinuity of $\{ f^{(k)} : f\in B\}$.

But a normed space has the Montel-Heine-Borel property if and only if it is finite-dimensional.