Natural flyers use soft wings to seamlessly enable a wide range of flight behaviours, including agile manoeuvres, squeezing through narrow passageways, and withstanding collisions. In contrast, conventional quadrotor designs rely on rigid frames that support agile flight but inherently limit collision resilience and squeezability, thereby constraining flight capabilities in cluttered environments. Inspired by the anisotropic stiffness and distributed mass-energy structures observed in biological organisms, we introduce FlexiQuad, a soft-frame quadrotor design approach that limits this trade-off. We demonstrate a 405-gram FlexiQuad prototype, three orders of magnitude more compliant than conventional quadrotors, yet capable of acrobatic manoeuvres with peak speeds above 80 km/h and linear and angular accelerations exceeding 3 g and 300 rad/s$^2$, respectively. Analysis demonstrates it can replicate accelerations of rigid counterparts up to a thrust-to-weight ratio of 8. Simultaneously, FlexiQuad exhibits fourfold higher collision resilience, surviving frontal impacts at 5 m/s without damage and reducing destabilising forces in glancing collisions by a factor of 39. Its frame can fully compress, enabling flight through gaps as narrow as 70% of its nominal width. Our analysis identifies an optimal structural softness range, from 0.006 to 0.77 N/mm, comparable to that of natural flyers' wings, whereby agility, squeezability, and collision resilience are jointly achieved for FlexiQuad models from 20 to 3000 grams. FlexiQuad expands hovering drone capabilities in complex environments, enabling robust physical interactions without compromising flight performance.