  We present the first detection of a correlation between the Lyman-$\alpha$
forest and cosmic microwave background (CMB) lensing. For each Lyman-$\alpha$
forest in SDSS-III/BOSS DR12, we correlate the one-dimensional power spectrum
with the CMB lensing convergence on the same line of sight from Planck. This
measurement constitutes a position-dependent power spectrum, or a squeezed
bispectrum, and quantifies the non-linear response of the Lyman-$\alpha$ forest
power spectrum to a large-scale overdensity. The signal is measured at
5~$\sigma$ and is consistent with the $\Lambda$CDM expectation. We measure the
linear bias of the Lyman-$\alpha$ forest with respect to the dark matter
distribution, and constrain a combination of non-linear terms including the
non-linear bias. This new observable provides a consistency check for the
Lyman-$\alpha$ forest as a large-scale structure probe and tests our
understanding of the relation between intergalactic gas and dark matter. In the
future, it could be used to test hydrodynamical simulations and calibrate the
relation between the Lyman-$\alpha$ forest and dark matter.
