In this work we present results of a direct search for strange quark matter (SQM) in cosmic rays with the PAMELA space spectrometer. If this state of matter exists it may be present in cosmic rays as particles, called strangelets, having a high density and an anomalously high mass-to-charge (A/Z) ratio. A direct search in space is complementary to those from ground-based spectrometers. Furthermore, it has the advantage of being potentially capable of directly identifying these particles, without any assumption on their interaction model with Earth's atmosphere and the long-term stability in terrestrial and lunar rocks. In the rigidity range from 1.0 to ∼1.0×10^{3} GV, no such particles were found in the data collected by PAMELA between 2006 and 2009. An upper limit on the strangelet flux in cosmic rays was therefore set for particles with charge 1≤Z≤8 and mass 4≤A≤1.2×10^{5}. This limit as a function of mass and as a function of magnetic rigidity allows us to constrain models of SQM production and propagation in the Galaxy.