We report the mechanical behavior of vertically aligned carbon nanotube films, grown on Si substrates using atmospheric pressure chemical vapor deposition, subjected to in situ large displacement (up to 70 μm) flat-punch indentations. We observed three distinct regimes in their indentation stress–strain curves: (i) a short elastic regime, followed by (ii) a sudden instability, which resulted in a substantial rapid displacement burst manifested by an instantaneous vertical shearing of the material directly underneath the indenter tip by as much as 30 μm, and (iii) a positively sloped plateau for displacements between 10 and 70 μm. In situ nanomechanical indentation experiments revealed that the shear strain was accommodated by an array of coiled carbon nanotube “microrollers,” providing a low-friction path for the vertical displacement. Mechanical response and concurrent deformation morphologies are discussed in the foam-like deformation framework with a particular emphasis on boundary conditions