Context-sensitive grammars without permutation rules Permutation rules are called those which are of the form $AB\Rightarrow^*BA$1. It is also proven that permutation rules expand context-free grammars and allow them produce non-context-free languages.

My question is: are any examples of context-sensitive grammars, which doesn't contain any permutation rules but still able to produce non-context-free languages?

1) Book, R.V. International Journal of Computer and Information Sciences (1973) 2: 129. doi:10.1007/BF00976059. More generally permutation rules have form of $Y_1 \dots Y_m\rightarrow Y_{\pi_1}\dots Y_{\pi_m}$, where $\pi$ is some permutation of $\{1\dots m\}$.
You have the Koruda normal form in mind. It is indeed often useful, in particular in simulation LBAs. 

However, many definitions of CSGs (including Chomsky's) are different; they require rules to have the form

$\qquad \alpha A \beta \to \alpha \gamma \beta$.

You can see how this precludes "permutation rules".

Obtain a concrete grammar by applying the simulation you find in a proof that KNF is equivalent to the classical definition.