A nonlocal derivative nonlinear Schrödinger (DNLS) equation is analytically studied in this paper. By constructing Darboux transformations (DTs) of arbitrary order, new determinant solutions of the nonlocal DNLS equation in the form of Wronskian-type are derived from both zero and nonzero seed solutions. Periodic solitons are obtained with different parameter choices. When one eigenvalue tends to another one, generalized DTs are constructed, leading to rogue waves. Due to complex parametric constraints, the derived solutions may have singularities. Despite this, the work presented in this paper can still provide a valuable reference for the study of nonlocal integrable systems.