Of the few counterexamples to this theorem that I was able to find, including this one, none of them involves an incomplete prehilbertian space and a closed incomplete convex subset.
I have another example with the space of square integrable sequences and the subspace of finitely supported sequences which again, is not closed.
I'm in search of an example of an incomplete prehilbertian space containing a closed convex subset which doesn't define a projector (hence the subset must be incomplete).
I naively tried considering the prehilbertian space of finitely supported sequences with the $l^2$ inner product and my closed convex subset was those sequences that start with a $0$. But even though the completeness assumption is not met, the projection function is still well-defined since all you have to do to project an element is replace its first term with a $0$.