The main purpose of this paper is to establish sharp Trudinger–Moser type inequalities on hyperbolic spaces for functions whose hyperbolic gradient is in the Lorentz space , . Namely, if is bounded and , we will show that there exists a constant such that for all with , we have where and is the hyperbolic gradient; if , we use only the norm rather than (unlike the case of Euclidean spaces) and will show that there exists a constant such that for all with , we have where