Gotten is an American word. It is used as the past-particle of get.

So, yes, it is appropriate and correct to say it, though the way you've used it sounds incorrect. Consider:

Sarah has gotten to liking him.

Also, this is not really used outside of the US. In the UK (and most everywhere else) you would say:

Sarah got to liking him.

Gotten is an American word. It is used as the past-participle of get.

So, yes, it is appropriate and correct to say it, though the way you've used it sounds incorrect. Consider:

Sarah has gotten to liking him.

Also, this is not really used outside of the US. In the UK (and most everywhere else) you would say:

Sarah got to liking him.