The Blakers-Massey element in $\pi_6(S^3)\cong\mathbb{Z}_{12}$ can be represented by such a map. This is done explicitly on page 3 of the paper https://arxiv.org/abs/math/0501091.
Let $\mathbb{H}$ denote the quaternions, and represent the $6$-sphere as $$ S^6 = \{(p,w)\in \mathbb{H}\times\mathbb{H} \mid \mathfrak{Re}(p)=0\mbox{ and } |p|^2+|w^2|=1\}. $$ The map $b:S^6\to S^3\subseteq \mathbb{H}$ is given by $$ b(p,w) = \frac{w}{|w|} e^{\pi p} \frac{\bar w}{|w|}, $$ where $e^{\pi p} = \cos(\pi |p|) + \sin(\pi|p|) \dfrac{p}{|p|}$ is the quaternionic exponential.
The fact that $b(-p,-w)=\overline{b(p,w)}$ is easily checked (and is noted in the proof of Theorem 1 in the linked paper).