The dimension of the real continuous functions as a vector space over $\mathbb{R}$ is not countable?
Solution 1:
The functions $f_t(x) = e^{tx}$, $t \in \mathbf R$ are linearly independent over $\mathbf R$. Can you prove it?
(Hint: suppose you have a minimal linear dependence relation among them, and use its derivative to produce a combination that is yet minimaler.)
Remark: we don't need the axiom of choice for this.
Solution 2:
Indeed $C([a,b],{\mathbb R})$ does not have countably-infinite dimension: no infinite-dimensional Banach space does (e.g. by the Baire category theorem).