We introduce a family of implicit integration methods for the dynamical low-rank approximation: the alternating-implicit dynamically orthogonal Runge-Kutta (ai-DORK) schemes. Explicit integration often requires restrictively small time steps and has stability issues; our implicit schemes eliminate these concerns in the low-rank setting. We incorporate our alternating iterative low-rank linear solver into high-order Runge-Kutta methods, creating accurate and stable schemes for a variety of previously intractable problems including stiff systems. Fully implicit and implicit-explicit (IMEX) ai-DORK are derived, and we perform a stability analysis on both. The schemes may be made rank-adaptative and can handle ill-conditioned systems. To evaluate nonlinearities effectively, we propose a local/piecewise polynomial approximation with adaptive clustering, and on-the-fly reclustering may be performed efficiently in the coefficient space. We demonstrate the ai-DORK schemes and our local nonlinear approximation technique on an ill-conditioned matrix differential equation, a stiff, two-dimensional viscous Burgers’ equation, the nonlinear, stochastic ray equations, the nonlinear, stochastic Hamilton-Jacobi-Bellman PDE for time-optimal path planning, and the parabolic wave equation with low-rank domain decomposition in Massachusetts Bay.