The simple answer is "no." This is easily demonstrated by looking at the sky at night. It is dark, and therefore the number of stars is finite. If there were an infinite number of stars, light would be streaming in to the Earth from all point in the sky, which would make the sky very (lethally) bright.
Further evidence stems from the cosmic microwave background radiation, which is highly uniform in all directions (isotropic and homogeneous), and about 3 degrees above absolute zero, which is completely inconsistent with an infinite number of stars in the universe.
Third is the finite volume and energy of the universe, where "universe" here refers to all that can be observed from Earth. This volume is defined by the speed of light combined with the time since the Big Bang. It is clearly finite based on strong observational evidence and theoretical work, which in turns precludes an infinite volume or energy in the universe.
Finally, current cosmology strongly favors a multiverse with a multitude of "pocket universes", of which ours is one. This view is based on the idea of eternal inflation as developed by Alan Guth and others. It should not be confused with the at present speculative notion of a cosmic landscape as found in string theory.
Instead, the notion of a vast and possibly infinite number of pocket universes is a consequence of current theory, which in turn is based on good observational evidence. That said, each of these pocket universes is separated from all others, so light from one cannot reach another. As above, this scenarios also precludes an infinite number of stars.
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