Q:

What part-of-speech is "running" in this example?

What part-of-speech is "running" in the following example?

The car comes with daytime running lights.

Is it a noun, verb (gerung), adjective, or something else?

A:

https://en.wikipedia.org/wiki/Daytime_running_lamp
Technically, it's a lamp (=lights) indicating a state, and that lamp is for usage at daytime.
'Daytime' acts as an adjective here, like 'night' does in the combination 'night bomber' where night can't be an object, since it's not about 'bombing the night', it's about night usage. That adjective applies to the combination 'running lights' where lights is definitely a noun. Unlike 'daytime' the gerund 'running' there functions like a noun, since the lights don't run and just indicate the car engine's state name called 'running'. That name could be any noun phrase. So the word in question is a gerund in function of a noun.

