"I like mint."
It can only be used verbally. Even if you are in a context that you do not need to care about politeness, だ is needed to make it grammatically correct.
Also it would be a wise choice to start with polite form first.
What makes this sentence different from others that allow you to drop the です?
There are two types of adjectives;
い adjectives that end in い are able to conjugate like verbs and do not require a copula to be a complete sentence. (高い、小さい、etc.)
な adjectives are adjectives that require a な to connect directly to a noun and cannot conjugate on their own so must be used with a copula.
Despite how it sounds, 好き is a な adjective and requires です or informally だ to be a complete sentence.
I don't understand why は isn't acceptable here. "As for mint, I like it" sounds like the right meaning to me.
It depends very much on the context. Usually when you tell somebody which your favourite is, there are supposed to be a few or more choices. In this example mint is one of them. Until you specifically mention mint, it is still just one of the choices, not a main topic. Using が actually distinguishes mint from the other choices. The hidden topic is "Me".
(As for me,) mint (rather than other flavours) is desirable.
Of course if mint has become a topic already, e.g. the previous sentence is a suggestion to buy mint flavor ice-cream, then your sentence would fit.