Here's an algebraic method that reduces the $n \geq 2$ case to the $n=1$ case, which is relatively easy. Useless bonus: you can replace $\mathbb{R}$ with any uncountable field (for suitable notions of "ball").
For each hyperplane in the collection, we can take the unique normal line through the origin. We get a pair of points on the unit hypersphere by taking the intersection with this line. If we collect all of the points arising from the hyperplanes, the coordinates generate a field extension $K$ of $\mathbb{Q}$ that has at most countably infinite transcendence degree (and is therefore countable and not all of $\mathbb{R}$).
Choose a point $x$ on the unit hypersphere whose first $n-1$ coordinates are algebraically independent of $K$ and of each other, and consider the line $\ell$ passing through $x$ and the origin. Each hyperplane can intersect $\ell$ in at most one point, since $x$ has nonzero inner product with the normal vector of any hyperplane in the collection.