Let's say I release a video where I say "sponsored by Company C" even though they never sponsored me.

Are there legal implications to lying about that, or is this action protected by the First Amendment?

At first, I thought that mentioning Company C is like doing free advertising for them, so why would they care. However, if Company C doesn't want to be associated with me, and the alleged association damages their reputation, then maybe this counts as [slander](https://dictionary.law.com/Default.aspx?selected=1969):

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> oral defamation, in which someone tells one or more persons an untruth about another, which untruth will harm the reputation of the person defamed.
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Since in this case, I wouldn't be blatantly saying a bad thing about the company (e.g. "their CEO is a murderer"), I imagine they would need to do a bit more work to prove to a court that their reputation was damaged.

(this is also assuming that slander applies to companies)