  We investigate Cu-doped Ba(Fe$_{1-x}$Cu$_x$)$_2$As$_2$ with transport,
magnetic susceptibility, and elastic neutron scattering measurements. In the
heavily Cu-doped regime where long-range stripe-type antiferromagnetic order in
BaFe$_2$As$_2$ is suppressed, Ba(Fe$_{1-x}$Cu$_x$)$_2$As$_2$ (0.145 $\leq x
\leq$ 0.553) samples exhibit spin-glass-like behavior in magnetic
susceptibility and insulating-like temperature dependence in electrical
transport. Using elastic neutron scattering, we find stripe-type short-range
magnetic order in the spin-glass region identified by susceptibility
measurements. The persistence of short-range magnetic order over a large doping
range in Ba(Fe$_{1-x}$Cu$_x$)$_2$As$_2$ likely arises from local arrangements
of Fe and Cu that favor magnetic order, with Cu acting as vacancies relieving
magnetic frustration and degeneracy. These results indicate locally broken
four-fold rotational symmetry, suggesting that stripe-type magnetism is
ubiquitous in iron pnictides.
