Arrowsmith was educated at Cheltenham Ladies College, read history at the University of Cambridge, and then read Social Science at the University of Liverpool and at Ohio University as a US-UK Fulbright Scholar. She is a co-founder of the Campaign for Nuclear Disarmament.
She has served eleven prison sentences for her political activities. In 1961 she was the subject of parliamentary questions after she was force-fed while on hunger strike in Gateside prison.
She stood against the then Prime Minister, James Callaghan, in his constituency of Cardiff South-East in the General election of 1979. She carried on sustained heckling during Callaghan's customary acceptance speech on re-election as an MP. With his typically avuncular nature Callaghan remarked that it was the first time he had "conducted a duet in returning a vote of thanks, and that it was not a particularly tuneful duet" . He then invited Arrowsmith to take the platform, which she did, while he, his supporters and all of the other candidates left the hall en masse.
- "Book Review of 'Somewhere Like This' by TrashFiction.com". Trashfiction.co.uk. Retrieved 2010-07-14.
- Goff, Hannah (2004-04-07). "UK | Peace campaigners return to Aldermaston". BBC News. Retrieved 2010-07-14.
- "Hansard 24 October 1961". Hansard.millbanksystems.com. 1961-10-24. Retrieved 2010-07-14.
- Coverage of election result on BBC Decision 79
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