Is the equivalence principle strictly fulfilled by general relativity?

The equivalence principle states

The outcome of any local experiment in a freely falling laboratory is independent of the velocity of the laboratory and its location in spacetime.

Any real local experiment needs some finite space, since nobody is a point particle himself. In principle I will always feel some deviation.

Since in (pseudo-)Riemannian geometry I can only change to flat coordnates in a single point $p$ of the spacetime manifold (and not in a finite neighborhood), is the deduction right, that the equivalence princpil is only fulfilled by general relativity in an approximative sense?