We realize a mechanical analogue of the Dicke model, achieved by coupling the spin of individual neutral atoms to their quantized motion in an optical trapping potential. The atomic spin states play the role of the electronic states of the atomic ensemble considered in the Dicke model, and the in-trap motional states of the atoms correspond to the states of the electromagnetic field mode. The coupling between spin and motion is induced by an inherent polarization gradient of the trapping light fields, which leads to a spatially varying vector light shift. We experimentally show that our system reaches the ultra-strong coupling regime, i.e., we obtain a coupling strength which is a significant fraction of the trap frequency. Moreover, with the help of an additional light field, we demonstrate the in-situ tuning of the coupling strength. Beyond its fundamental interest, the demonstrated one-to-one mapping between the physics of optically trapped cold atoms and the Dicke model paves the way for implementing protocols and applications that exploit extreme coupling strengths. The quantum Rabi model (QRM) describes the interaction of a two-level emitter with a single quantized mode of the electromagnetic field. Together with its extension for an ensemble of emitters, i.e., the Dicke model (DM), it constitutes a cornerstone of quantum optics [1]. The physics predicted by the QRM and the DM strongly depends on the relative values of the mode frequency, ω, and the coupling strength between the two-level system (TLS) and the bosonic mode, g. For weak coupling, i.e., g/ω 1, the rotating wave approximation (RWA) applies. In this case, the QRM and the DM reduce to the Jaynes-Cummings and the Tavis-Cummings models, respectively. The RWA breaks down in the ultra-strong coupling regime (USC), i.e., for g/ω 0.1. When increasing the coupling strength further, one enters the deep-strong coupling regime (DSC) [2]. For such high values of g/ω, new phenomena are expected [3-7]. The existence of a quantum phase transition in the thermo-dynamic limit adds to the richness of the DM [8-10]. Furthermore, USC and DSC may enable novel protocols for quantum communication and quantum information processing [11-13]. Over the last decade, USC was reached using various experimental platforms [14-22]. More recently, DSC was achieved in circuit quantum electrodynamics [23, 24] as well as by coupling a THz metamaterial with cyclotron resonances in a two-dimensional electron gas [25]. While these systems reach record-high ratios of g/ω, the large coupling strengths make state preparation and read-out challenging. For this reason, alternative routes were proposed to achieve large coupling in experimental platforms that, at the same time, offer a high level of control and tunability. Following this path, the QRM in the USC and DSC regimes was simulated using circuit quantum elec-trodynamics [26, 27], and DSC was studied with single trapped ions [28]. Here, we implement a mechanical analogue of the Dicke model by coupling the spin of individual neutral atoms to their quantized motion in a trapping potential. In our approach, the coupling is enabled by spatial gradients of the vector light shift inherent to optical mi-crotraps. Fluorescence spectroscopy, which was recently used to measure the temperature of atoms in a nanofiber-based trap after degenerate Raman cooling [29], grants access to the energy spectrum of the system. We observe vacuum Rabi splittings and transitions between dressed states that both clearly and consistently reveal an ultra-strong spin-motion coupling in our experiment, i.e., the coupling strength is a significant fraction of the mode frequency. Furthermore, we demonstrate that the coupling strength can be readily tuned in situ using an additional laser light field.