In this talk, we present several results concerning strongly subdifferentiable (SSD, for short) points in Banach spaces and revisit this classical yet somewhat overlooked differentiability notion. We first motivate the concept, positioning it between Gâteaux and Fréchet differentiability and highlighting its geometric significance. We then examine structural aspects of norms exhibiting SSD points, with particular emphasis on Lipschitz-free spaces. In this setting, we provide conditions ensuring that elementary molecules are SSD points and introduce metric perturbation techniques yielding abundance of SSD elements. As an application, we show that for uniformly discrete metric spaces one can construct bi-Lipschitz perturbations making SSD points dense. This is a joint work with Christian Cobollo, Petr Hájek and Mingu Jung.