We study a class of acoustic metamaterials formed by layers of perforated plates and producing negative refraction and backward propagation of sound. A slab of such material is shown to act as a perfect acoustic lens, yielding images with subwavelength resolution over large distances. Our study constitutes a nontrivial extension of similar concepts from optics to acoustics, capable of sustaining negative refraction over extended angular ranges, with potential application to enhanced imaging for medical and detection purposes, acoustofluidics, and sonochemistry.