Could there be a type of mass that moves in a direction perpendicular
to its momentum?
I think a similar phenomenon may be possible, but it would need to be modified: Momentum that is perpendicular to the direction of movement of the mass
I'm not sure it's actually possible - but of the answers I've read here so far, these things seem to overlook or ignore an answer based on real physics: electromagnetism
Consider that there exists a force in electromagnetism where the force is perpendicular to the direction of movement/momentum. When electrons move, magnet fields are generated perpendicular to their direction of motion. There is a significant body of research on this, it's well known and proven. Do all the research you like.
So, if energy and momentum or mass and momentum... some similar combination... were in some way related like electricity and magnetism, then it may be that "momentum" could be perpendicular to the movement of the mass, like magnetism is perpendicular to the movement of electrons.
This may not be what you were asking, but I just thought you would want an answer that provided the possibility of something similar to what you asked. I hope this is helpful.
*I will mention that I majored in physics, and this sounds ridiculous, but this is a site about "world building" and what might be possible, not what can be proven.
I assumed electromagnetism was sufficiently easy to research that I didn't need citations, but since a comment was made, I will provide a reference to a wiki: https://en.wikipedia.org/wiki/Magnetic_field#Magnetic_field_due_to_moving_charges_and_electric_currents
And note that in the wiki, it discusses the "Lorentz Force":
F = q v X B
This is fairly analogous to the force equation:
F = m a = m v / t
Note that the cross product in the Lorentz Force means that the force is perpendicular to the velocity vector. However, this is constant in time because it describes the interaction of a quantity of charge (q) with a given velocity (v).
For your question, the problem is determining what this "field" is that would interact with the "momentum of mass" perpendicular to it's direction of movement. I provide this as an analogy - it may exist, and we may be like scientist were over a hundred years ago when they were first learning about the relationship between electricity and magnetism. But.. probably not. It's just a curious parallel to me.