That would be a wild thing to accomplish, given that neutrinos are created at relativistic speeds, have virtually zero mass, and don’t react with anything but gravity and the weak force.
Whatever gravity “trap” you make is going to pale in comparison to the gravity wells around us (earth, the sun, etc) and the weak force scales exponentially with energy. So a slow-moving neutrino would interact even less with the weak force than a relativistic one.
Probably! And even potentially some orbiting the planet. They’d be part of the cosmic neutrino background, neutrinos around since the big bang which have had their velocity red-shifted to Newtonian speeds.
But they interact with things even less than relativistic neutrinos. Even a light-year of lead wouldn’t be enough to register a hit from one.
And even if you did have light-years of super dense detector material lying around, the energy deposited by such a neutrino is so tiny that you wouldn't be able to register the hit.
Speed is relative, so if you accelerated away from the sun at 0.999999c hoping to hang out in a slow-moving stream of solar neutrinos, you’d remove the energy needed for them to interact with the weak force.
They’d be all around you, but you would have exponentially smaller chances of interacting with them.
Whatever gravity “trap” you make is going to pale in comparison to the gravity wells around us (earth, the sun, etc) and the weak force scales exponentially with energy. So a slow-moving neutrino would interact even less with the weak force than a relativistic one.