Welcome to Bizarro Physics Department! Kindly step this way we may have an answer for your interesting problem. Firstly, throw away all notions of conventional and rational physics. It's time to turn the dial up to "bizarre"!
Assume the planets all stationary relative to each other because spacetime and gravitation operates differently. Instead being associated specifically and exclusively with mass gravitation is a property of spacetime itself. The general gradient of gravitation across spacetime is flat. This means gravitation is effectively zero everywhere. Except there are exceptions to this rule and exceptions aplenty.
There will be numerous spikes of gravitational potential at points in the fabric of spacetime. Please note these spikes are stationary relative to each other. At the locations of these spikes of gravitational potential this is where planets will form and they will remain relatively stationary to each other.
Effectively gravitation will be a localized phenomena. If we assume the gravitation emerging in spikes from spacetime will only attract the mass constituting the mass incorporated in planets. While the gravitational mass of the planetary matter does not attract the gravitational mass of planets. Effectively the planets do not gravitational attract each other, the gravitational mass in any planet is attracted to the gravitation in the sharp gravitational gradients of the spacetime spikes. The planets won't attract each other, so the system won't collapse together. This universe will resemble an assemblage of stationary planets.
Any mass at a distance from the spacetime spikes won't be attracted. This means other planets won't be pulled together. Nor spaceships be attracted. This means if a spacecraft is at relative rest with respect to the system of planets it won't be moving and it won't fall, move or be attracted to any of the planets.
The bizarro cosmology outlined in this answer may satisfy all the criteria sought by the OP or something may have been left that renders this conceptual contrivance nonsense. If so, this can be done in the comments.
A side-note: there is nothing to prevent planets rotating on their axes.