The dynamics of combustion waves propagating in a solid reacting material is investigated in the presence of radiative heat losses within the one-step Arrhenius reaction mechanism and the narrow sample approximation. As expected, a C-shaped response curve caused by the competition of heat release and heat dissipation is found where the lower branch of the response curve is unstable. Along the upper branch, the period doubling Feigenbaum's cascade occurs with increasing values of heat-loss intensity leading to chaotic dynamics. Finally, for the further increase of the heat-loss parameter, the persistent chaotic dynamics is replaced by the sudden extinction. It must be emphasized that this extinction occurs for significantly lower heat losses than the steady extinction limit associated with the turning point of the C-shaped response curve. The most unexpected finding is that the temporal interval of chaotic behavior preceding the sudden extinction event may be extremely long with apparently unpredictable duration. These results may be relevant for combustion processes in solid energetic materials or other similar systems. (C) 2020 The Combustion Institute. Published by Elsevier Inc. All rights reserved.