In this paper, we classify the irreducible representations of the trigonometric Cherednik algebras of rank 1 in characteristic p > 0 . There are two cases. One is the “quantum” case, where “Planck’s constant” is nonzero and generic irreducible representations have dimension 2 p . In this case, smaller representations exist if and only if the “coupling constant” k is in F p ; namely, if k is an even integer such that 0 ≤ k ≤ p − 1 , then there exist irreducible representations of dimensions p − k and p + k , and if k is an odd integer such that 1 ≤ k ≤ p − 2 , then there exist irreducible representations of dimensions k and 2 p − k . The other case is the “classical” case, where “Planck’s constant” is zero and generic irreducible representations have dimension 2. In that case, one-dimensional representations exist if and only if the “coupling constant” k is zero.