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We study the tidal response of a superfluid neutron star in a binary system, focussing on Newtonian models with superfluid neutrons present throughout the star's core and the inner crust. Within the two-fluid formalism, we consider the main aspects that arise from the presence of different regions inside the star, with particular focus on the various interfaces. Having established the relevant theory, we determine the tidal excitation of the most relevant oscillation modes during binary inspiral. Our results suggest that superfluid physics has a negligible impact on the static tidal deformation. The overwhelming contribution to the Love number is given by, as for normal matter stars, the ordinary fundamental mode (f-mode). Strong entrainment, here described by a phenomenological expression which mimics the large effective neutron mass expected at the bottom of the crust, is shown to have significant impact on the superfluid modes, but our results for the dynamical tide are nevertheless similar to the static limit: the fundamental modes are the ones most significantly excited by the tidal interaction, with the ordinary f-mode dominating the superfluid one. We also discuss the strain built up in the star's crust during binary inspiral, showing that the superfluid f-mode may (depending on entrainment) reach the limit where the crust breaks, although it does so after the ordinary f-mode. Overall, our results suggest that the presence of superfluidity may be difficult to establish from binary neutron star gravitational-wave signals.