Regular or flavored chewing gum is forbidden according to all sources I found but there are some permitting sugar-free flavorless gum which is apparently considered as a non-food then.
Many poskim forbid chewing gum without distinction, e.g., Ohel Moshe Vol. 2-113, Yalkut Yosef, Hillel Omer OC 278
R Eli Mansour cites R Ovadia Yossef forbidding regular chewing gum
Hacham Ovadia Yosef in his work Yabia Omer (vol. 7, Orah
Haim 33) rules that if one chews gum coated with sugar
or other sweetening agent, he must recite the Beracha of She’hakol. By
the same token, it would be forbidden to chew sweet chewing gum on a
The reference to sweet chewing gum is because R Ovadia Yossef permits flavorless gum and is not concerned by excessive saliva production
When it comes to chewing gum on a fast day, Hacham Ovadia rules that
if the gum has a sweet flavor (like most chewing gum does), then it
may not be chewed on a fast day. Flavorless gum may be chewed, even it
has the effect of increasing the amount of saliva in the mouth, since
saliva is considered part of the body and it may be swallowed on a
R Eli Mansour brings another source permitting chewing gum more broadly
Hacham Ovadia Hedaya (1890-1969), in his work Yaskil Abdi, rules that
chewing gum is not considered “eating,” and thus one does not recite a
Beracha before he chews gum.
Yom Kippur (and Tisha bAv?) likely have stricter laws, see here for instance
Please CYLOR before chewing !
Note that most poskim believe a chewing gum needs a sheakol bracha with possible exceptions again for sugar-free gum.