We study the size of the sets of gradients of bump functions on the Hilbert space $\ell _2$, and the related question as to how small the set of tangent hyperplanes to a smooth bounded starlike body in $\ell _2$ can be. We find that those sets can be quite small. On the one hand, the usual norm of the Hilbert space $\ell _2$ can be uniformly approximated by $C^1$ smooth Lipschitz functions $\psi$ so that the cones generated by the ranges of its derivatives $\psi '(\ell _2)$ have empty interior. This implies that there are $C^1$ smooth Lipschitz bumps in $\ell _2$ so that the cones generated by their sets of gradients have empty interior. On the other hand, we construct $C^1$-smooth bounded starlike bodies $A\subset \ell _2$, which approximate the unit ball, so that the cones generated by the hyperplanes which are tangent to $A$ have empty interior as well. We also explain why this is the best answer to the above questions that one can expect.