Context. The mass discrepancy in the Universe has not been solved by the cold dark matter (CDM) or the modified Newtonian dynamics (MOND) paradigms so far. The problems and solutions of either scenario are mutually exclusive on large and small scales. It has recently been proposed, by assuming that dark matter is a superfluid, that MOND-like effects can be achieved on small scales whilst preserving the success of ΛCDM on large scales. Detailed models within this “superfluid dark matter” (SfDM) paradigm are yet to be constructed.Aims. Here, we aim to provide the first set of spherical models of galaxy clusters in the context of SfDM. We aim to determine whether the superfluid formulation is indeed sufficient to explain the mass discrepancy in galaxy clusters.Methods. The SfDM model is defined by two parameters. Λ can be thought of as a mass scale in the Lagrangian of the scalar field that effectively describes the phonons, and it acts as a coupling constant between the phonons and baryons. m is the mass of the DM particles. Based on these parameters, we outline the theoretical structure of the superfluid core and the surrounding “normal-phase” dark halo of quasi-particles. The latter are thought to encompass the largest part of galaxy clusters. Here, we set the SfDM transition at the radius where the density and pressure of the superfluid and normal phase coincide, neglecting the effect of phonons in the superfluid core. We then apply the formalism to a sample of galaxy clusters, and directly compare the SfDM predicted mass profiles to data.Results. We find that the superfluid formulation can reproduce the X-ray dynamical mass profile of clusters reasonably well, but with a slight under-prediction of the gravity in the central regions. This might be partly related to our neglecting of the effect of phonons in these regions. Two normal-phase halo profiles are tested, and it is found that clusters are better defined by a normal-phase halo resembling an Navarro-Frenk-White-like structure than an isothermal profile.Conclusions. In this first exploratory work on the topic, we conclude that depending on the amount of baryons present in the central galaxy and on the actual effect of phonons in the inner regions, this superfluid formulation could be successful in describing galaxy clusters. In the future, our model could be made more realistic by exploring non-sphericity and a more realistic SfDM to normal phase transition. The main result of this study is an estimate of the order of magnitude of the theory parameters for the superfluid formalism to be reasonably consistent with clusters. These values will have to be compared to the true values needed in galaxies.Key words: X-rays: galaxies: clusters / dark matter / galaxies: kinematics and dynamics / galaxies: clusters: general / cosmology: miscellaneous