I'm now at a startup, so no (director == founding exec until you reach a certain size). But yes, for some definition of "lead" which involves managing a bunch of people as well, and director, which involves managing a bunch of managers, that's something that happens.

Small team tech-only leads who are not managing people will not typically be promoted to director levels at most big companies, but if you're actually managing people and your skip-manager loves you, it's totally possible even if your team is small.