We apply a Lyapunov function to obtain conditions for the existence and uniqueness of small classical time-periodic solutions to first-order quasilinear 1D hyperbolic systems with (nonlinear) nonlocal boundary conditions in a strip. The boundary conditions cover different types of reflections from the boundary and integral operators with delays. In the first step, we use a Lyapunov approach to derive sufficient conditions for the robust exponential stability of the boundary value problems for a linear(ised) homogeneous problem. Under those conditions and a number of non-resonant conditions, in the second step, we prove the existence and uniqueness of smooth time-periodic solutions to the corresponding linear inhomogeneous problems. In the third step, we prove a perturbation theorem stating that the periodic solutions survive under small perturbations of all coefficients in the hyperbolic system. In the last step, we apply the linear results to construct small and smooth time-periodic solutions of the quasilinear problems.