I just realize that my question is actually rather trivial the way I posed it: There exist non-commutative semisimple examples (with the $2\times 2$ matrix algebra being the smallest example). There also exist commutative non-semisimple examples, the smallest one given by
$$e_0\cdot e_0 = e_0,\quad e_0\cdot e_1=e_1,\\
e_1\cdot e_0=e_1,\quad \mu(e_1)=1\;,$$
where $\mu$ is the linear form generating the symmetric non-degenerate pairing and all other entries $0$.
All one needs to do is consider direct sums or tensor products of a non-commutative (but semisimple) and a non-semisimple (but commutative) example, and it will be both non-commutative and non-semisimple. The smallest such example is of dimension $6$.
I guess a more sensible but a bit artificial question to ask would have been whether there are examples that do not come from the tensor product or direct sum of a commutative and a semi-simple example.