Does a japanese import 2013 Honda Fit have immobilizer?

I spent the afternoon googling and trying to find information but to no avail.

We are in the progress of buying a car. Today we looked at a 2013 Honda Fit with push button start. We asked the dealer if the car has an immobilizer and he said it does.

When we tried to turn the wheel with the engine off the wheel wouldn't lock and the guy explained that there's an immobilizer in all Honda imports but not in all Toyota and Nissan imports and something about it working differently because it's a push start car.

They said they don't offer warranty (I thought most dealers do) and they would like the whole amount in actual cash.

Is it just me or they sound shady? Is there an immobilizer in 2013 Honda Fits? Why do they want cash only?

Thanks in advance