Thinking about this matter, assuming Whittaker did leave with Chibnall, couldn't that potentially damage her career? It'd be one thing if BBC were unfairly forcing Chibnall out against his will, in that case I can see showing solidarity with your producer would make you look better to other potential producers you'd work with later on. But, according to these "reports" Chibnall is voluntarily leaving, in which case if Whittaker chose to leave the show just because of a change of producers, wouldn't others be hesitant to cast her since there's every bit the possibility she'd leave again due to change in producers?
Of course, there's too much wrong with this story. BBC didn't want RTD to leave after five years, and tried to make him stay by backing a dump truck loaded with cash to his doorstep. When he refused that, they considered ending the show. Moffat also tried to call it quits after five years, and they persuaded him to stay two more years to produce one more season. I highly doubt BBC would let Chibnall after only one successful season that has arguably reinvigorated the show.
Also, Whittaker seems very happy with the show and appears to be enjoying herself. Also, she knows how important many of audience in general, but most especially the female audience consider having a female Doctor to be. As a self-described feminist (according to her Wikipedia page) I do not see why she would leave a job she enjoys where she is able to be an inspiration and positive role model to other women/girls (and even men/boys of course) just because a man is leaving. That just doesn't add up at all.
