Is the distance between two elements of a normed vector space at least the distance between their projections on the unit ball?

Not true in general, as balls are not necessarily round. Take for instance $X=\mathbb R^2 $ with $$\|x\|_\infty=\max\{|x_1|,|x_2|\}. $$ Choose $$x=(1,3/4), \ \ \ y =(1/4,5/4) $$ Then $\|x\|=1$, and $$\|x-y\|=3/4,\ \ \ \|x-y/\|y\|\|=4/5 $$

The answer is negative. The operator norms of the two matrices below are equal to $1$: $$ A=\pmatrix{1&0\\ 0&0},\ B=(\sqrt{2}-1)\pmatrix{2&1\\ 1&0}. $$ However, numerically we have $\|A-B\|=0.50879>0.50406=\|A-\frac{17}{16}B\|$. So, a counterexample is exhibited by putting $(x,y)=(A,\frac{17}{16}B)$.