She is driving me crazy.

She will talk to me but none of the chat lines lead to me dismissing her.

Reminds me of my sister, just send her off to toil in a field somewhere.

If she is bugged, try picking up another companion. This will force her to go away.

When you walk up to her, initiate, and one of the options other than talk should say Part Ways or something.

Spectacle Island.

My version of Alcatraz...

There is a dimiss option after you pick "talk"

Piper is amazing.

Under the "Talk" options with her you should get "Dismiss". If that doesn't appear, she'll leave you if you murder someone. Make her hate you enough and she'll aggro on you, I think, for you to kill.

On Xbox One :

Command/Interact (A Button) > Talk (Y Button) > Dismiss (X Button) > Choose destination (Up/Down on LS or D-Pad and A Button to confirm, or cancel with the B Button to have her return to Publick Occurrences in Diamond City).

Now that is a statement I can agree with.

I often have her along with me, and I even fully romanced her.

Choose another companion and then when asked to send her home if you really don't want her around choose a settlement you rarely visit.