I think it is "possible", but I think it is a moot question...
Now some people feel that the source material is not relevant, because they feel the TV show-only viewers should not have to rely on it. --Which is true, but that is not the same as either there being more there for the person whom has also read the books and/or the possibility that despite the condensing of the source material, the TV series is still playing with the same things to meet a very similar end, at least, philosophically. It should also be noted that neither the books or the tv series are finished to gain full absolution of truths and also that Martin has stated he will not fully explain his metaphysics, meaning that (in both media) there will be mysteries left to ponder about.
For instance, The Dragon Pit scene featured in the final episode of season 7, will have more meaning to a book reader, than a TV-only viewer. A few episodes before, the TV series discusses and name drops Aegon the Conquer. The reason this is done, is because there is a correlation between Dany and her three dragons juxtaposing Aegon and his three dragons when he conquered Westoros and built the Dragon Pitt for his dragons, but book readers will get an additional thrill when seeing Dany arrive on Drogan, because the books suggest that Dany's three dragons are the literal reincarnates of Aegon's dragons. So Drogon being in the dragon pit is really like Balor the Dread arriving home. This is thematic and plays into the cycle cosmology story we have, including kinds of metaphysical transformations the series highlights, including "living-dead" characters, such as Jon Snow being resurrected with viewers also learning (in this episode) that he was officially named Aegon...
The reason I went into all that, is because it ties into my arguement, as TFM also play into Metaphysical transformations--including using something "dead" and making it "alive" and how this plays to "muddied" identity or the exploration of identity (in Philosophy this called Mereology)
In the show we can assume that A Faceless Man is either somewhat telekinetic or can receive sensory information from another person. 'Jaqen H'gar' says, "The Girl has many names on her lips..." and then names them accurately back to Arya. How can he know this, unless he has mind-reading capabilities?
Now the books will refer this other Faceless Man we meet in Braavos as "The Kindly Man", but the TV show does give him another name to establish that he is different from Jaqen, despite that he assumes Jaqen's face. The TV series does try to establish this though by introducing the character with a different unknown face. He seems to know and/or value Arya's previous experiences, because assuming Jaqen's face is a clear indication of this. So transference of some sort again ensues. (I believe the books go further though and make it more apparent that they can also retrieve memories from the faces themselves).
Then Early on when Arya fails, she sees herself dead and then "transfers" herself into a near identical body, but she is now blind. So again we do not know if "Arya" in some way actually "died" during this magical process, but she still was consciously the same Arya, but it could another instance of transference...
But when we get to the fight with the Waif (and remember, the Waif is a Faceless Man too, whom is assuming an identity that would/is meant to "antagonize" and "provoke" Arya to force her forward--that the waif's face is already that of someone who died), the viewers do not see it, to know it's outcome.
Arya then returns seemingly with the Waifs face (remember, when Arya becomes blind there are "two" Arya's in that scene), and puts it [back] in the hall. The Kindly Man essentially gives her approval and tells her she is No One" (A Faceless Man), but Arya says, she is Arya and she is going home [to Westoros]. The Kindly Man gives a faint smile, which is unclear if he is amused for it's own sake, or because he knows something else...
In Season seven we get a lot of strong indications that this is still IN PART Arya, because she continues on her killing spree (Arya's previous motive) and because when she learns about her family from Pot Pie she changes her course to more literally go home. Then she runs across Nymeria whom eventually recognizes her...
However, when one looks closer at the situation there are a lot of signs that tell us that, in some way, is not the same Arya that existed prior to fighting the Waif. She is fearless, is better at social improve, and has skill sets she was never able to master before that battle. She also seems to begin to read Sansa's mind when she questions Sansa's choice of their parents' bedroom and her need for creature comforts...
So I think it is safe to say that Arya did go through a metaphysical transformation, but that it is unclear if the process actually killed her previous self, if the waif stole her face (and has her memories) or not...
I think it probably has to be moot question at this point, because whether or not this girl was or wasn't Arya (if Arya died when she became blind or the waif is wearing her face), it's clear that the realization of Arya's of family being alive and beginning to reunite, caused a change of planned course of action in season 7---and it shows an attachment more for meaningful relationships, then either revenge or debt.
So I think even if Arya died and this person wasn't Arya prior to leaving Braavos, she may be becoming Arya anyways. If we looks at the series from it's bigger picture, then we know "fate" also plays a role, as we have characters that are either implied to be reincarnates and/or are metaphysically transforming in order to be here. Jon Snow is one prime example of a character, whom may be in part The Prince Who Was Promised, and was not suppose to be "Dead" during this era. It might be interesting to entertain the idea if The Waid did steal Arya's face, if it was possible that Arya's soul "reprinted" back onto her/it, because maybe no matter what "Arya" [transformed] is suppose to be here. But it's possible that TFM are not done with Arya and there could be further conflict between her and them that has yet to be resolved.
Examples of Metaphysical Transformations:
- Bran Warging - Experiencing what it is like to become an animal (a direwolf)
- Bran - A Greenseer - Time Traveling as an Observer to present and past events
- Bran - The Tree-Eyed Raven - Advanced Greenseer whom has collective conscious or collective memory --looses himself to memory, history, time traveling.
- The Knight King - Human Turned White Walker
- Hodors Paradox - Two Points in time "change" character.
- Army of Living Dead (bodies used to serve White Walkers)
- Wights - People serving a bigger force (Beric Donderion and Jon Snow are said to be Wights of Light, as Beric mentions one looses parts of themselves each Resurrection)
- Lady Stoneheart
- Reincarnation- Dragons, Legendary Figures (Age of Heroes)
Other Identity Related Ideas:
- Theon emasculation (Theon Deon Argument?)
- Sansa - Bad Experience Made Her Stronger
- Jyne Pole - Impersonates Sansa
- Young Griff - Claiming to be Aegon Targeryen
Dany is kinetic with her Dragons and is not in control of her own behavior. Not the same as warging, but similar, albeit not as clean cut.