What is the Irish for this?
Focloir.ie has rith le d'anam.

What about: rith i bpéin an anama?

Rith i dtánaiste d’anama = Run for your dear life

Féach FGB: tánaiste: Rith sé i dtánaiste a anama.
Alternative versions in FGB: Rith sé lena anam; rith sé i mbarr, i dtánaiste, ar theann, a anama, he ran for his life.

I’ve come across this idiom reading this book: Imtheachtaí an Oireachtais / Madra na nOcht gCos ⁊ scéalta eile
There it is with other verbs and with article instead of possessive:

⁊ is aṁlaiḋ a ḃí an garsún ag teaċt i dtánaiste an anama ⁊ …
Do ċomáin an c’lúr leis i dtánaiste an anama ⁊ …
⁊ cú ’n-a ḋiaiḋ i dtánaiste an anama ...
in its foclóir:
Tánaiste (the next thing to ?) ; i dt. an anama, (lit. in the next thing to the life or soul ?), as hard as he could run

corpas.focloir.ie (again with article)
ag féachaint ar an mbabliac ag rith i dtánaiste an anama
Tar éis ' bheith ag rith i dtánaiste an anama ar feadh dhá uair a ' chloig
(both: Pinocchio, P. Ó Buachalla)

Thank you. I like the quote from Pinocchio, apart from the apostrophe before bheith.