The Skyrme model from nuclear physics is a quasilinear modification of the nonlinear sigma model (wave maps). The modification is designed to break scale invariance and, in three spatial dimensions, it is known to prevent the formation of singularities in contrast to the nonlinear sigma model which suffers self-similar collapse. It is not known, however, whether or not this modification has the same effect for higher dimensions, as heuristics based on the conserved energy suggest that the dominating contribution to blow-up dynamics should come from the quasilinear term. To that end, I study the strong-field Skyrme model which ignores the sigma model term and retains the quasilinear term. Remarkably, the strong-field model possesses an explicit self-similar solution which develops a singularity in finite time. I will discuss recent progress toward establishing the stability of this explicit solution and its potential implications for the development of singularities of the Skyrme model in higher dimensions. Our emphasis will be on novel techniques developed to treat derivative-nonlinear terms in the stability analysis. Such terms arise from the quasilinear nature of the strong-field Skyrme model, and do not appear in previous analyses of the stability of self-similar blow-up for semilinear waves.