  We continue studying $6D, {\cal N}=(1,1)$ supersymmetric Yang-Mills (SYM)
theory in the ${\cal N}=(1,0)$ harmonic superspace formulation. Using the
superfield background field method we explore the two-loop divergencies of the
effective action in the gauge multiplet sector. It is explicitly demonstrated
that among four two-loop background-field dependent supergraphs contributing to
the effective action, only one diverges off shell. It is also shown that the
divergences are proportional to the superfield classical equations of motion
and hence vanish on shell. Besides, we have analyzed a possible structure of
the two-loop divergences on general gauge and hypermultiplet background.
