Yes, if we consider normed spaces $X$ to be vector spaces over $\mathbb{R}$ or $\mathbb{C}$. In that case, such spaces are path-connected: for $x , y \in X$ define $f(t) = (1-t)x + ty$ from $[0,1]$ to $X$, which is well-defined in that case as $[0,1] \subseteq \mathbb{R} \subseteq \mathbb{C}$ and continuous in the topology from the norm metric, and is a path from $x$ to $y$.