Claim that anyone who believes that white privilege exists is a white supremacist by definition.

She'll reject that notion.

So, take her through this logic train:

Ask her when the exact date was when white people gained their white privilege.

If not an exact date, then a time period.

Seriously, ask her what historical point in time white people suddenly had every advantage over virtually every other race of peoples simply by virtue of their skin tone.

Tangentially ask why every other race failed to take the place of white people as the privileged race. What makes white people so special that they alone are guilty of a superior racial privilege? What allowed white people to advance beyond all other races and seat themselves at the top of the social hierarchy? What causes white people to be universally oppressive of other races?

Then ask her why other races failed... What made them less capable of securing the lofty spot at the top?

There's the stinger.

The only way she can get out of saying that white people are just naturally better at being the oppressive ruling class (proving she is a white supremacist) is to say that there is no white privilege, there is only cultural privilege, of which race is merely a variable, just like anything else that makes two people different.

Of that, I'm sure, we can all agree.

Ultimately, individuals have no privilege; individual privilege is a fantasy. Privilege is not inherent to an individual's circumstances. When stranded on a remote island, what privilege could an individual possibly have? How does their skin tone entice a coconut to fall off a tree easier?

Privilege only applies in a context -- in societal cultures and how the majority view individuals based on stereotypes unique to the cultural world view.

No amount of self-hate is going to change the cultural privilege dynamic. Only positive representation of your unique attributes, including race, will encourage others to view those like you positively. It's really as simple as that, and is a fundamental component of human psychology.

So, ask her, how has hating her own race and thinking down upon herself for actions she was never involved with, done by people she isn't related to, and before she was ever born, changed her life, or anyone else's life, for the better?

Has hating herself been a benefit in any way? Has it encouraged her to stay positive and pursue her dreams? Or has it convinced her she is unworthy of happiness solely on account of her skin color?

Has her own self deprecation set a positive look for those she represents? Or has it only served to reinforce the stereotypes surrounding those like her?

There is no good in hating one's self for things you can not and couldn't have ever controlled.