High mutation rates in non-recombining populations invoke several processes that may lead to extinction. R.A. Fisher argued that an intermediate mutation rate would be optimal, ensuring a steady input of beneficial mutations while avoiding the detrimental accumulation of deleterious mutations. Here, we utilized a time-serial experimental evolution framework, combined with novel population genetic inference methods, to investigate the process of mutational meltdown and the role of error catastrophe in influenza A virus (IAV) populations with artificially increased mutation rates. A novel antiviral drug favipiravir, which increases the genome-wide mutation rate in IAV, was administered with varying dosage strategies. Under low concentation regimes, we report the first evidence to date for the ability of virus populations to adapt to favipiravir. However, under high concentration regimes, we observe extinction in all replicates. We discuss the observed evolutionary dynamics with respect to previous theoretical results pertaining to mutational meltdown and evolutionary rescue.