An anvil-attachable layer for use with a surgical stapler, or fastening instrument, wherein a proximal end portion of the layer is attached to a staple cartridge assembly, for example. The layer may be attached to the staple cartridge assembly by an adhesive, weld, or a staple-cartridge-based clamp, wherein the attachment is weak enough to allow the layer to pull away from the staple cartridge assembly with stapled tissue. Alternatively, the layer can include two or more lateral slits that define a connector region that can be cut by a knife of a surgical stapler to release the layer.