Well, Archduchess Anastasia Nikolaevna certainly existed, but the fact of her death at the same time as the rest of her family has been proven. When the killers of the Romanov family disposed of the bodies, they ended up burying the Tsarevich and one of the daughters (it's disputed whether this was Maria or Anastasia) in a separate grave from the rest. This grave wasn't discovered until quite some time after the main grave was found, which (between discoveries) added fuel to the fire of the "Anastasia survived somehow" rumors which had been flying since not long after the family's murder. These rumors seem to have begun when false claimants appeared saying they were Anastasia. There was also a false claimant who said he was Tsarevich Alexei, but he didn't catch the attention of the public the way the fake Anastasias did. The most determined fake was one Anna Anderson, who claimed for decades to be Anastasia despite several people, who had known the real one, saying she couldn't be (sadly, Gleb Botkin, son of the also-murdered Romanov family doctor and a childhood friend of the archduchesses, was taken in by her and thought she really was his old friend.) Anderson was proven a liar by DNA testing after she died, and the discovery of the second grave proved beyond doubt that the entire family died together.

If you'd like to learn more about the real Anastasia and her siblings, I'd recommend The Romanov Sisters by Helen Rappaport.

To clarifying, are you asking if there is any evidence that Anastasia survived?