We study the dependence of galaxy clustering on luminosity and stellar mass at redshifts z ~ [0.2-1] using the first zCOSMOS 10K sample. We measure the redshift-space correlation functions xi(rp,pi) and its projection wp(rp) for sub-samples covering different luminosity, mass and redshift ranges. We quantify in detail the observational selection biases and we check our covariance and error estimate techniques using ensembles of semi-analytic mock catalogues. We finally compare our measurements to the cosmological model predictions from the mock surveys. At odds with other measurements, we find a weak dependence of galaxy clustering on luminosity in all redshift bins explored. A mild dependence on stellar mass is instead observed. At z~0.7, wp(rp) shows strong excess power on large scales. We interpret this as produced by large-scale structure dominating the survey volume and extending preferentially in direction perpendicular to the line-of-sight. We do not see any significant evolution with redshift of the amplitude of clustering for bright and/or massive galaxies. The clustering measured in the zCOSMOS data at 0.5<z<1 for galaxies with log(M/M_odot)>=10 is only marginally consistent with predictions from the mock surveys. On scales larger than ~2 h^-1 Mpc, the observed clustering amplitude is compatible only with ~1% of the mocks. Thus, if the power spectrum of matter is LCDM with standard normalization and the bias has no unnatural scale-dependence, this result indicates that COSMOS has picked up a particularly rare, ~2-3 sigma positive fluctuation in a volume of ~10^6 h^-1 Mpc^3. These findings underline the need for larger surveys of the z~1 Universe to appropriately characterize the level of structure at this epoch.
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