Water under vinyl flooring

My rabbits live in a shed with vinyl flooring. I pulled it partially up to clean and there is a large amount of water underneath. I've pulled it up a few days to try and dry it out, but the next day there is always a bit more and obviously with the current cold weather, it's not drying quickly. I don't know if this is rainwater which has got under when the door is left open to their run (we've not had rain for over a week but it could previously have saturated the wooden floor), condensation or damp rising up through the wooden floor.

Has anyone had a similar problem, and if so, how did you solve it? I have installed a cat flap so I won't need to leave the door open in the day once they've learned how to use it, so that should limit the rain that gets in. I could possibly remove some of the vinyl, except around their toilet trays. Unfortunately we've had rats living under the shed which means I've had to block off around the shed to stop those getting in, so there's only very limited air flow under it.

Could you build a false floor under the vinyl, just lay some thin strips of wood and then either wire mesh (probably my preference) or wood on top before putting the vinyl back down? I can't see how it is ever going to dry out enough otherwise. Longterm you need to work out where it is coming from to really deal with it though, and that is likely to mean creating an airflow underneath that is rat proof. Airbricks perhaps, or metal mesh.

Thank you Dollyanna. Metal mesh for the air flow is a good idea, I'd be worried the rats would get through the airbrick holes - I've looked online and there are long strips of mesh which I think are used in plastering, which would do the job to block of the sides of the shed from the rats and also let in airflow, so I will look at getting some of that - and also lifting the vinyl as you suggest in the meantime to try and dry the wood. The scraps of wood that I've kept as they might come in handy one day might actually come in handy for this!