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Using analytical theory and hybrid-kinetic numerical simulations, we demonstrate that, in a collisionless plasma, long-wavelength ion-acoustic waves (IAWs) with amplitudes $δn/n_0 \gtrsim 2/β$ (where $β\gg{1}$ is the ratio of thermal to magnetic pressure) generate sufficient pressure anisotropy to destabilize the plasma to firehose and mirror instabilities. These kinetic instabilities grow rapidly to reduce the pressure anisotropy by pitch-angle scattering and trapping particles, respectively, thereby impeding the maintenance of Landau resonances that enable such waves' otherwise potent collisionless damping. The result is wave dynamics that evince a weakly collisional plasma: the ion distribution function is near-Maxwellian, the field-parallel flow of heat resembles its Braginskii form (except in regions where large-amplitude magnetic mirrors strongly suppress particle transport), and the relations between various thermodynamic quantities are more `fluid-like' than kinetic. A nonlinear fluctuation-dissipation relation for self-sustaining IAWs is obtained by solving a plasma-kinetic Langevin problem, which demonstrates suppressed damping, enhanced fluctuation levels, and weakly collisional thermodynamics when IAWs with $δn/n_0 \gtrsim 2/β$ are stochastically driven. We investigate how our results depend upon the scale separation between the wavelength of the IAW and the Larmor radius of the ions, and discuss briefly their implications for our understanding of turbulence and transport in the solar wind and the intracluster medium of galaxy clusters.