We investigate Cu-doped Ba(Fe1−xCux)2As2 with transport, magnetic susceptibility, and elastic neutron scattering measurements. In the heavily Cu-doped regime where long-range stripe-type antiferromagnetic order in BaFe2As2 is suppressed, Ba(Fe1−xCux)2As2 (0.145≤x≤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(Fe1−xCux)2As2 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 fourfold rotational symmetry, suggesting that stripe-type magnetism is ubiquitous in iron pnictides.