  We analyze strongly lensed images in 8 galaxy clusters to measure their dark
matter density profiles in the radial region between 10 kpc and 150 kpc, and
use this to constrain the self-interaction cross section of dark matter (DM)
particles. We infer the mass profiles of the central DM halos, bright central
galaxies, key member galaxies, and DM subhalos for the member galaxies for all
8 clusters using the Qlens code. The inferred DM halo surface densities are fit
to a self-interacting dark matter (SIDM) model, which allows us to constrain
the self-interaction cross section over mass $\sigma$/m. When our full method
is applied to mock data generated from two clusters in the Illustris-TNG
simulation, we find results consistent with no dark matter self-interactions as
expected. For the eight observed clusters with average relative velocities of
$1458_{-81}^{+80}$ km/s, we infer $\sigma$/m = $0.082_{-0.021}^{+0.027}$
cm$^2$/g and $\sigma$/m < 0.13 cm$^2$/g at the 95% confidence level.
