A remarkable feature of layered transition-metal oxides—most famously, the high-temperature superconductors—is that they can display hugely anisotropic electrical and optical properties (for example, seeming to be insulating perpendicular to the layers and metallic within them), even when prepared as bulk three-dimensional single crystals. This is the phenomenon of 'confinement', a concept at odds with the conventional theory of solids, and recognized1 as due to magnetic and electron–lattice interactions within the layers that must be overcome at a substantial energy cost if electrons are to be transferred between layers. The associated energy gap, or 'pseudogap', is particularly obvious in experiments where charge is moved perpendicular to the planes, most notably scanning tunnelling microscopy2 and polarized infrared spectroscopy3. Here, using the same experimental tools, we show that there is a second family of transition-metal oxides—the layered manganites La2-2xSr1+2xMn2O7—with even more extreme confinement and pseudogap effects. The data demonstrate quantitatively that because the charge carriers are attached to polarons (lattice- and spin-textures within the planes), it is as difficult to remove them from the planes through vacuum-tunnelling into a conventional metallic tip, as it is for them to move between Mn-rich layers within the material itself.