Is every unambiguous grammar regular?

While searching for an answer to this question I found out that there is an unambiguous grammar for every regular language. But is there a regular language for every unambiguous grammar? How can I prove that this is/isn't true?

• Have you tried finding an unambiguous grammar for $a^nb^n$? – greybeard Sep 16 '20 at 6:58

The following grammar is unambiguous yet generates a non-regular language: $$S \to aSb \mid \epsilon$$
• As a side note, this works for every $a,b \in \Sigma$ where $a \neq b$. A very notable example would be the language of matched parenthesis $(^n)^n$. – Polygnome Sep 15 '20 at 22:53