We exploit the high spatial resolution and high cadence of the Interface Region Imaging Spectrograph (IRIS) to investigate the response of the transition region and chromosphere to energy deposition during a small flare. Simultaneous observations from the Reuven Ramaty High Energy Solar Spectroscopic Imager provide constraints on the energetic electrons precipitating into the flare footpoints, while observations of the X-Ray Telescope, Atmospheric Imaging Assembly, and Extreme Ultraviolet Imaging Spectrometer (EIS) allow us to measure the temperatures and emission measures from the resulting flare loops. We find clear evidence for heating over an extended period on the spatial scale of a single IRIS pixel. During the impulsive phase of this event, the intensities in each pixel for the Si IV 1402.770 Å, C II 1334.535 Å, Mg II 2796.354 Å, and O I 1355.598 Å emission lines are characterized by numerous small-scale bursts typically lasting 60 s or less. Redshifts are observed in Si IV, C II, and Mg II during the impulsive phase. Mg II shows redshifts during the bursts and stationary emission at other times. The Si IV and C II profiles, in contrast, are observed to be redshifted at all times during the impulsive phase. These persistent redshifts are a challenge for one-dimensional hydrodynamic models, which predict only short-duration downflows in response to impulsive heating. We conjecture that energy is being released on many small-scale filaments with a power-law distribution of heating rates.