Is 'punishing' a verb

Answer 1

Yes and no!

The word looks like the present participle of a verb, punish, and you can certainly use it that way:

If the teacher keeps the whole class back, she will be punishing everyone for something only one person did.

But present participles can take on other roles. One is to be a noun:

Punishing people always makes me uncomfortable.

Another is as an adjective:

That is the most punishing stage of the Tour de France.

More than two thousand years ago, the Romans appreciated this versatility of verb participles. In their highly inflected language, Latin, both present and past participles were inflected in the same manner as words that were unequivocally adjectives.

So the answer to the question has to be: It depends on the context. The role that the word punishing plays has to be determined from the rest of the sentence, or even the rest of the discourse.