Thin low-frequency acoustic absorbers that are economical to produce using a large-scale manufacturing process are scarce, and their efficiency is often limited to a narrow frequency range. In this paper, perforated gypsum foams are shown to achieve high absorption levels for layers thinner than 1/15 of a wavelength. With a mass density of 150 kg/m³, they are as practical in use as conventional porous absorbers. To reach acoustic absorption levels higher than 0.7, perforations with a diameter smaller than 1 mm open a fraction of the initially closed pores. To allow additional design freedom, the foams' pore size can be varied between 1 and 3 mm, while keeping the wall thickness as low as 0.1 mm. Simulations of the microscopic fluid flow in a representative volume element show how the combination of foam properties and perforation patterns can be combined such that sub-wavelength absorption is obtained. Comparing the predicted sound absorption with impedance tube measurements demonstrates that the solution can yield a wideband low-frequency sound absorption peak. The most remarkable example exhibits an absorption peak higher than 0.7 between 525 Hz and 875 Hz for a layer of 25 mm, a result equivalent to 1/21 of the incident sound wavelength.