Irreducible elements are non-units by definition. That is, they are non-units because the definition explicitly requires them to be; if it didn't, they could be units.

Can a unit be irreducible?

An irreducible is an element of an integral domain that cannot be factored without one of its factor being a unit. For example when the integral domain in question are the integers, the units are -1 and 1 and the irreducibles are the primes.

What does it mean for an element to be irreducible?

An element of a ring which is nonzero, not a unit, and whose only divisors are the trivial ones (i.e., the units and the products , where is a unit).