  We study the dynamical thermal conductivity of the Kitaev spin model on a
two-leg ladder. In contrast to conventional integrable one-dimensional spin
systems, we show that heat transport is completely dissipative. This is a
direct consequence of fractionalization of spins into mobile Majorana matter
and a static $Z_{2}$ gauge field, which acts as an emergent thermally activated
disorder. Our finding rests on three complementary calculations of the current
correlation function, comprising a phenomenological mean-field treatment of
thermal gauge fluctuations, a complete summation over all gauge sectors, as
well as exact diagonalization of the original spin model. The results will also
be contrasted against the conductivity discarding gauge fluctuations.
