Yes. Section 43 of the federal trademark law (the Lanham Act) prohibits false or misleading representations that are “likely to cause confusion, or to cause mistake, or to deceive as to the affiliation, connection, or association of such person with another person, or as to the origin, sponsorship, or approval of his or her goods, services, or commercial activities by another person.” Section 43 is worded broadly enough to reach many forms of false or misleading advertising practices. A false statement or suggestion that a company or a person (such as a celebrity) endorses, sponsors or is associated with a company, product or service could violate the Lanham Act, whether or not it also amounts to common law trademark infringement. This kind of conduct can also give rise to state-law causes of action for unfair competition, appropriation of personality, fraud, etc.
Nominative fair use, by the way, is subject to the condition of truthfulness. For example, saying that the Trump Organization endorses your company’s cryptocurrency products would be nominative use of the TRUMP trademark. You do not get the benefit of the nominative fair use defense, however, if the statement that the products are endorsed by the Trump Organization is not true.