  We present the first spatially resolved (~0.5 kpc) measurements of the
molecular gas depletion time $\tau_{depl}$ across the disk of the interacting
spiral galaxy NGC\,2276, a system with an asymmetric morphology in various SFR
tracers. To estimate $\tau_{depl}$, we use new NOEMA observations of the
$^{12}$CO(1-0) emission tracing the bulk molecular gas reservoir in NGC 2276,
and extinction-corrected H$\alpha$ measurements obtained with the PMAS/PPaK
integral field unit for robust estimates of the SFR. We find a systematic
decrease in $\tau_{depl}$ of 1-1.5 dex across the disk of NGC 2276, with a
further, abrupt drop in $\tau_{depl}$ of ~1 dex along the galaxy's western
edge. The global $\tau_{depl}$ in NGC 2776 is $\tau_{depl}=0.55$ Gyr, insistent
with literature measurements for the nearby galaxy population. Such a large
range in $\tau_{depl}$ on sub-kpc scales has never previously been observed
within an individual isolated or pre-merger system. When using a
metallicity-dependent molecular gas conversion factor X$\rm_{CO}$ the variation
decreases by 0.5 dex. We attribute the variation in $\tau_{depl}$ to the
influence of galactic-scale tidal forces and ram pressure on NGC 2276's
molecular interstellar medium (ISM). Our observations add to the growing body
of numerical and observational evidence that galaxy-galaxy interactions
significantly modify the molecular gas properties and star-forming activity
within galactic disks throughout the interaction, and not just during the final
merger phase.
