The matrix $M = \left(\begin{array}{clcr} 1&1&0&0\\1&1&1&0\\0&1&1&0\\0&0&0&1\end{array} \right)$ is an example where the dimension of the space in question is odd.

The matrix $M = \left(\begin{array}{clcr} 1&1&0&0\\1&1&1&0\\0&1&1&0\\0&0&0&1\end{array} \right)$ is an example where the dimension of the space in question is odd.

(Later edit: However if $M = M^{t}$ and also $M \in {\rm Sp}(2n,2),$ then we in fact have $M^{2} = I_{2n}$ as Darij Grinberg and Noam Elkies implicitly noted in comments).

The matrix $M = \left(\begin{array}{clcr} 1&1&0&0\\1&1&1&0\\0&1&1&0\\0&0&0&1\end{array} \right)$ is an example where the dimension of the space in question is odd.

The matrix $M = \left(\begin{array}{clcr} 1&1&0&0\\1&1&1&0\\0&1&1&0\\0&0&0&1\end{array} \right)$ is an example where the dimension of the space in question is odd.