  Let $K$ be a field and let $\sigma$ be an automorphism and let $\delta$ be a
$\sigma$-derivation of $K$. Then we show that the multiplicative group of
nonzero elements of the division ring $D=K(x;\sigma,\delta)$ contains a free
non-cyclic subgroup unless $D$ is commutative, answering a special case of a
conjecture of Lichtman. As an application, we show that division algebras
formed by taking the Goldie ring of quotients of group algebras of torsion-free
non-abelian solvable-by-finite groups always contain free non-cyclic subgroups.
