Let $X$ be an uncountable discrete space with a distinguished element $0$. We view the product space $X^X$ as the space of maps $\phi: X \to X$. The set
$$ E := \{ \phi \in X^X: \phi(\phi(0)) = 0 \}$$
is easily seen to be clopen, hence the indicator function $1_E: X \to {\bf R}$ is continuous, but depends on all of the (uncountably many) coordinates of $X^X$.

The key point here (which was inspired by Nate's comment based on the earlier incorrect attempt at solving this problem) is that deciding whether a given map $\phi$ belongs to $E$ requires only a finite number of (adaptive) evaluations of $\phi$, but the set of (non-adaptive) locations where $\phi$ could potentially need to be evaluated is uncountable.

Note that a similar construction works for $X \times \{0,1\}^X$ using the set $E := \{ (x, \phi) \in X \times \{0,1\}^X: \phi(x)=0\}$; thus even a single highly non-compact factor is enough to generate a counterexample. (But I am not sure what happens if one insists that all of the factors be sigma-compact, in particular can one construct a continuous function $f: {\bf N}^{\bf R} \to {\bf R}$ that depends on uncountably many coordinates?)