Lipschitz mappings, contingents, and differentiability

The main purpose of the paper is to show that, for each real normed space $Y$ of infinite dimension, each number $L>0$, and each at most countable set $Q\subset\mathbb{R}$, there exists a Lipschitz mapping $f\colon\mathbb{R}\to Y$, with constant $L$, whose graph has a tangent everywhere, whereas $?$ is not differentiable at any point of $Q$.