To date, more than 750 planets have been discovered orbiting stars other than the Sun. Two sub-classes of these exoplanets, "hot Jupiters" and their less massive counterparts "hot Neptunes," provide a unique opportunity to study the extended atmospheres of planets outside of our solar system. We describe here the first far-ultraviolet transit study of a hot Neptune, specifically GJ436b, for which we use HST/STIS Lyman-$α$ spectra to measure stellar flux as a function of time, observing variations due to absorption from the planetary atmosphere during transit. This analysis permits us to derive information about atmospheric extent, mass-loss rate from the planet, and interactions between the star and planet. We observe an evolution of the Lyman-$α$ lightcurve with a transit depth of GJ436b from $8.8±4.5\%$ near mid-transit, to $22.9±3.9\%$ $∼2$ hours after the nominal geometric egress of the planet. Using data from the time-tag mode and considering astrophysical noise from stellar variability, we calculate a post-egress occultation of $23.7±4.5\%$, demonstrating that the signature is statistically significant and of greater amplitude than can be attributed to stellar fluctuations alone. The extended egress absorption indicates the probable existence of a comet-like tail trailing the exoplanet. We calculate a mass-loss rate for GJ436b in the range of $3.7\times10^6 -1.1\times10^{9}$ g s$^{-1}$, corresponding to an atmospheric lifetime of $4\times10^{11}-2\times10^{14}$ years.