  It has been proposed that gravothermal collapse due to dark matter
self-interactions (i.e. self-interacting dark matter, SIDM) can explain the
observed diversity of the Milky Way (MW) satellites' central dynamical masses.
We investigate the process behind this hypothesis using an $N$-body simulation
of a MW-analogue halo with velocity dependent self-interacting dark matter
(vdSIDM) in which the low velocity self-scattering cross-section,
$\sigma_{T}/m_{x}$, reaches 100 cm$^{2}$g$^{-1}$; we dub this model the vd100
model. We compare the results of this simulation to simulations of the same
halo that employ different dark models, including cold dark matter (CDM) and
other, less extreme SIDM models. The masses of the vd100 haloes are very
similar to their CDM counterparts, but the values of their maximum circular
velocities, $V_{max}$, are significantly higher. We determine that these high
$V_{max}$ subhaloes were objects in the mass range [$5\times10^{6}$,
$1\times10^{8}$] $M_\odot$ at $z=1$ that undergo gravothermal core collapse.
These collapsed haloes have density profiles that are described by single power
laws down to the resolution limit of the simulation, and the inner slope of
this density profile is approximately $-3$. Resolving the ever decreasing
collapsed region is challenging, and tailored simulations will be required to
model the runaway instability accurately at scales $<1$ kpc.
