Are there cases when the word "best" as an adjective could be used without "the"?

Superlative adjectives are normally preceded by the, but for reasons of brevity and snappiness, headlines, slogans and banners often omit words that would occur in formal prose. That seems to be what is happening here.

Banner English is similar to headline English: it's telegraphic and filled with omissions because the point of the language is to communicate the key concepts as quickly and briefly as possible, not to produce a grammatically correct complete sentence. For the banner, the English is perfect. For a formal written sentence, you'd probably want to say These are the best {advertisement designs / designs from advertisements}. [I see that Barrie England and I agree 100% on this one.]

If the noun phrase modified by best follows best, then you normally use the article, e.g., "Here are the five choices. This is the best choice". If it doesn't, then you can omit the article, e.g., "Here are the five choices. This is {best / the best}".

"I am so nervous about asking a question on that tough ELU site. What is going to happen?

"Best case scenario, you will simply get a helpful answer. My best guess is that you may also get a bit of attitude."