To some extent, yes, it is possible: you take a heavy nucleus with $+q$ and attach one electron to it. It will be an ion with the total charge $+q-1$ and with Hydrogen-like orbital motions, except for being more relativistic and with the ground state still spherically symmetric. All the other configurations ("ring-like") are quasi-stable due to possibility to emit photons with getting into the "atomic" ground state. Read something about Hydrogen-like ions.
P.S. In principle, one can imagine a superposition of different states with different quantum numbers such that the wave packet will ressemble more or less localized "particle" moving around the nucleus (kind of a "coherent state"). But again, such state is only quasi-stationary since there is still interaction with photon field leading to decay such an atomic (ionic) state into another superposition with smaller atomic energies and with several photons flying away.