We explore the properties of chiral superfluid thin films coating a curved surface. Because of the vector nature of the order parameter, a geometric gauge field emerges and leads to a number of observable effects such as anomalous vortex-geometric interaction and curvature-induced mass and spin supercurrents. We apply our theory to several well-known phases of chiral superfluid He-3 and derive experimentally observable signatures. We further discuss the cases of flexible geometries where a soft surface can adapt itself to compensate for the strain from the chiral superfluid. The proposed interplay between geometry and chiral superfluid order provides a fascinating avenue to control and manipulate quantum states with strain.