I'm flying soon and have been given a maximum carry on bag dimension of 14 inches x 9 inches x 22 inches (23 x 35 x 56 cm), and the body of my luggage exactly fits those dimensions. If the handles exceed the limits, will my bag be allowed on the flight?
Short answer to the question in the topic: yes.
It may depend on the airline but e.g. EasyJet explicitly states "including handles and wheels" and I would say it is a general rule.
In your case it may be not allowed. Size may be checked at the gate by having you put your bag into a box/cage of those dimensions. Obviously, if the handle gets in a way, it won't fit in. But, if it looks like a standard, small carry-on suitcase, that is not bulging, I doubt they will bother with checking.