 Solution

For the centered ball, the Hahn-Banach theorem supplies a norm-one real functional with $f(y)=\Vert y\Vert >r$. Hence $f(z)\le r$ on the ball, and any $t\in (r,\Vert y\Vert )$ separates it strictly from $y$.

For the general center, apply the same construction to $y-x$: choose $\Vert f\Vert =1$ and $f(y-x)=\Vert y-x\Vert$. ThenFor example,gives $f(z)<t<f(y)$ for every $z\in B$. The strict gap uses exactly the assumption that $y$ lies outside the closed ball.