Yes, exactly what Douglas Keith said. The kinetic portion of the Hamiltonian is "non-local" in the position basis in exactly the same way that the potential portion of the Hamiltonian is non-local in the momentum basis: it appears in (powers of) the derivative.

If you want to talk about locality in terms of minimizing interactions between different basis states, then the basis is in eigenstates of the Hamiltonian, which is going to be neither position nor momentum.