It is not possible that $D^{\times}$ is a torsion group unless $D$ is a field.
Let $D$ be a division ring of characteristic $p$. Denote the center of $D$ by $Z(D)$.
Lemma 1: Every finite subgroup of $D^{\times}$ is commutative.
Proof: The $\mathbb F_q$-algebra generated by the subgroup is a finite subring. One easily sees that it has to be a subfield. Indeed, multiplication by non-zero elements is injective and hence surjective since the ring is finite. This implies that every non-zero element has a multiplicative inverse. We conclude that the subfield (and hence the subgroup) is commutative by Wedderburns theorem. q.e.d.
Herstein’s Lemma: Suppose $a$ is a noncentral, torsion element of $D^\times$. Then, there exists $y \in D^{\times}$ such that $yay^{−1} = a^i \neq a $ for some $i\geq 2$.
You can find a proof in "A First Course in Noncommutative Rings" by T.Y. Lam.
Theorem: Let $D$ be a division ring such that for any $a\neq 0$, there exists $n\geq 1$ such that $a^n = 1.$ Then, $D$ is a field.
Proof: Assume that $D$ is not a field. Let $0 \neq a \not \in Z(D)$. By Herstein's Lemma, there exists $y \in D^{\times}$ such that $yay^{-1} \neq a$. Since $y$ is torsion and normalizes the subgroup generated by $a$, $G=\langle a,y \rangle$ is a finite subgroup of $D^{\times}$. By Lemma 1 the group $G$ is commutative. But $ya \neq ay$ and we get a contradiction. q.e.d.
It is even enough to assume that all additive commutators are of finite order. This is sometimes called Jacobson’s commutativity theorem. A proof can also be found in the book by T.Y. Lam.
