The effect of intense X-ray laser interaction on argon clusters is studied theoretically with a mixed quantum/classical approach. In comparison to a single atom we find that ionization of the cluster is suppressed, which is in striking contrast to the observed behavior of rare-gas clusters in intense optical laser pulses. We have identified two effects responsible for this phenomenon: A high space charge of the cluster in combination with a small quiver amplitude and delocalization of electrons in the cluster. We elucidate their impact for different field strengths and cluster sizes. PACS numbers: 33.90.+h, 32.80.Hd, 36.40.Wa, 41.60.Cr The advent of femtosecond laser pulses has triggered numerous activities in the field of laser-matter interaction. In particular novel, non-linear processes in atoms induced by the available high intense fields from these ultrashort pulses have been a challenge for experiment and theory [1]. To understand more generally the energy transfer from intense laser light to matter, complex targets like atomic clusters have been studied [2,3,4,5]. These experiments received considerable attention due to the observed dramatic effects like emission of very fast ions [2] and electrons [3] or the production of coherent X-ray radiation [4]. So far such studies are almost exclusively restricted to visible or infrared wavelengths [6]. The new X-ray free electron laser (xfel) sources, under construction at DESY in Hamburg [7] and at the LCLS in Stanford [8], will change this situation. They can deliver intense laser pulses at high frequencies (from VUV to hard X-ray) and thus open a new regime of strong-field atomic physics.Here we present theoretical investigations of X-ray (frequencyhω = 350 eV, i. e. wavelength λ ≈ 3.5 nm) laser interaction with argon clusters at intensities of I ≈ 3.5·10 14 . . . 3.5·10 18 W/cm 2 . In this laser regime the interaction is notably different from long-wavelength pulses which is evident from the ponderomotive energy E pond ∼ I/ω 2 . It represents the average kinetic energy of a free electron in a laser field, while ∆x ∼ √ I/ω 2 is the quiver amplitude of the electron, i. e., the spatial excursion in the laser field. For the given laser parameters one finds E pond ≈ 0.4 meV . . . 4 eV, vastly different from E pond ≈ 20 eV . . . 200 keV for a 780-nm-laser at the same intensities. The small ponderomotive energy has two important consequences: (i) Despite the high intensities the laser-atom interaction is of non-relativistic and perturbative nature. The latter is also clear from the so-called Keldysh parameter γ which gives the ratio between the tunneling time and the laser period [9] and can be rewritten in terms of the binding energy E bind and the ponderomotive energy E pond as γ = (E bind /2E pond ) 1/2 . For the L and M -shells of argon with E bind = 326, 249, 29.3, 15.8 eV we always find γ > 1 and for the inner shells even γ ≫ 1. Obviously the laser period is far too small for field-or tunnel-ionization. Rather, ionization is due to single-photon absorption....