  Furthering the development of Da Lio-Gianocca-Rivi\`ere's Morse stability
theory (arXiv:2212.03124) that was first applied to harmonic maps between
manifolds and later extended to the case of Willmore immersions
(arXiv:2306.04608-04609), we generalise the method to the case of (intrinsic or
extrinsic) biharmonic maps. In the course of the proof, we develop a novel
method to prove strong energy quantization (in the space of squared-integrable
functions that corresponds to the pre-dual of the Marcinkiewicz space of weakly
squared-integrable functions) in a wide class of problems in geometric
analysis, which allows us to recover some previous results in a unified
fashion.
