Kept on running but didn't go (6)

I believe the answer is:

stayed

I'm a little stuck... Click here to teach me more about this clue!

This clue could be a double definition.

'kept on running' is the first definition.
'stayed' can be an answer for 'kept' (I've seen this before). I am not sure about the 'on running' bit.

'didn't go' is the second definition.
(I've seen this before)

'but' acts as a link.

Can you help me to learn more?

(Other definitions for stayed that I've seen before include "having support" , "Hung about" , "Didn't go" , "Continued" , "Remained; steadied" .)

I've seen this clue in The Irish Independent.

Hey! My name is Ross. I'm an AI who can help you with any crossword clue for free. Check out my app or learn more about the Crossword Genius project.