  We report observations of HD 80606 using the 10.4-m Gran Telescopio Canarias
(GTC) and the OSIRIS tunable filter imager. We acquired very-high-precision,
narrow-band photometry in four bandpasses around the K I absorption feature
during the January 2010 transit of HD 80606b and during out-of-transit
observations conducted in January and April of 2010. We obtained differential
photometric precisions of \sim 2.08e-4 for the in-transit flux ratio measured
at 769.91-nm, which probes the K I line core. We find no significant difference
in the in-transit flux ratio between observations at 768.76 and 769.91 nm. Yet,
we find a difference of \sim 8.09 \pm 2.88e-4 between these observations and
observations at a longer wavelength that probes the K I wing (777.36 nm). While
the presence of red noise in the transit data has a non-negligible effect on
the uncertainties in the flux ratio, the 777.36-769.91 nm colour during transit
shows no effects from red noise and also indicates a significant colour change,
with a mean value of \sim 8.99\pm0.62e-4. This large change in the colour is
equivalent to a \sim 4.2% change in the apparent planetary radius with
wavelength, which is much larger than the atmospheric scale height. This
implies the observations probed the atmosphere at very low pressures as well as
a dramatic change in the pressure at which the slant optical depth reaches
unity between \sim770 and 777 nm. We hypothesize that the excess absorption may
be due to K I in a high-speed wind being driven from the exoplanet's exosphere.
We discuss the viability of this and alternative interpretations, including
stellar limb darkening, starspots, and effects from Earth's atmosphere. We
strongly encourage follow-up observations of HD 80606b to confirm the signal
measured here. Finally, we discuss the future prospects for exoplanet
characterization using tunable filter spectrophotometry.
