Cat's nine lives ultimately running non-stop (9)
I believe the answer is:
incessant
'non-stop' is the definition.
(I've seen this before)
'cat's nine lives ultimately running' is the wordplay.
'ultimately' suggests the final letters.
'running' indicates an anagram.
The final letter of 'lives' is 's'.
'cats'+'nine'+'s'='catsnines'
'catsnines' with letters rearranged gives 'INCESSANT'.
(Other definitions for incessant that I've seen before include "Since Stan was continual." , "Persistent" , "Never-ending" , "Instances being continually repeated" , "Endless" .)