tl;dr: A simple argument seems to show that the sufficient statistics in a canonical exponential family are necessarily pairwise independent, which is clearly not true. So, what goes wrong in the derivation below?
Derivation (updated). This argument avoids reparametrization issues. Suppose we have an exponential family $\mathfrak{P}=\{p_{\theta}:\theta\in\Omega\}$ given by \begin{align} p_{\theta}(x) &= h(x)e^{\theta\cdot x - A(\theta)}, \quad x\in\mathbb{R}^{k}. \quad (*) \end{align}
Then obviously \begin{align*} p_{\theta}(x) &= h(x)e^{-A(\theta)}\prod_{i=1}^{k}e^{\theta_{i}x_{i}}. \end{align*}
We can assume $h(x)=1$ without any loss of generality (e.g. by absorbing $h(x)$ into the dominating measure). Reparametrizing in terms of the sufficient statistics $t$,
We have the factorization $p_{\theta}(x)=f_{1}(x_{1})\cdots f_{k}(x_{k})$ (e.g. let $f_1(x_1)=e^{-A(\theta)}e^{\theta_{1}x_{1}}$ and $f_i(x_i)=e^{\theta_{i}x_{i}}$ for $i>1$). This factorization is enough to ensure that $x_{1},\ldots,x_{k}$ are independent.
Thus, the sufficient statistics in a canonical exponential family are always mutually independent. What goes wrong in this argument?
Note. There always exists a dominating measure such that $(*)$ holds, see Propsition~1.5 here. The same author calls this a standard exponential family. For the curious, this question arose from trying to understand some of the arguments in this textbook.
(Update: @Henry below has a very simple counterexample that proves that this is definitely false.)