The formation process of binary stars and multiple systems is poorly understood. Here, we seek to determine the typical outcome of protostellar collapse and to constrain models of binary formation by core fragmentation during collapse, using high-resolution millimeter continuum imaging of very young (Class 0) protostars observed at the beginning of the main accretion phase. We carried out a pilot high-resolution study of 5 Class 0 objects, using the most extended (A) configuration of the IRAM PdBI at 1.3 mm, which allow us to probe the multiplicity of Class 0 protostars down to separations a ~50 AU and circumstellar mass ratios q ~0.07. We show that our PdBI observations revealed only wide (>1500 AU) protobinary systems and/or outflow-generated features. When combined with previous millimeter interferometric observations of Class 0 protostars, our pilot PdBI study tentatively suggests that the binary fraction in the ~ 75-1000 AU range increases from the Class 0 to the Class I stage. It also seems to argue against purely hydrodynamic models of binary star formation. We briefly discuss possible alternative scenarios to reconcile the low multiplicity rate of Class 0 protostars on small scales with the higher binary fraction observed at later evolutionary stages.