If you want a simple yes or no answer, then the answer has to be, "No." There are a number of things that can lead to a redshift in the light that a observing device receives relative to the light that is emitted by the source. Relative motion can do this. A difference in the gravitational field between emitter and source can do this. Relative time dilation can do this. The expansion of space can do this. The presence of dust between source and observer can make the intensity of light appear more red than previously; though this does not produce the same phenomena that we refer to as redshift, it is a potential source of error in observations of redshift. These are all things discussed in a decent textbook on cosmology. Probably in a good textbook on General Relativity (though they may omit a discussion of dust, since that is a more specialized subject). How do we have evidence of this? I always recommend the books of Clifford Will. "Was Einstein Right?" is a good book for a general audience, but he has a good textbook on GR that includes specific references to the evidence for GR as well as a nice section on the Parameterized Post-Newtonian framework used to test theories of gravity. Even though it's fairly old now, the basics are still relevant.