Can the lift fans used on a F-35 be driven by a turboshaft engine instead of the jet engine? If yes, which one?
There is no difference between turbofan, turboprop and turboshaft engine except for what is mounted on the shaft. There are many cores that are used in all of turbofan, turboprop and turboshaft engines. So yes, they can be driven by turboshaft. In a sense, the engine that drives them qualifies as turboshaft simply because it drives them.
That said, the lift fan of F-35 is absurdly inefficient. The thing is that lift is equal to mass flow through the fan/propeller times change of speed of the air, but the power required is equal to mass flow times square of change of speed, plus drag that grows with RPM.

Now the fan is small, so it has small mass flow and so it needs to accelerate the air to great speed and to do that it also needs high rpm, both greatly increasing the power required to spin it.

F-35 can live with that, because it only spends very short time in hover and a lot of time at high speed, so it can trade efficiency at hover for low drag at speed. But anything that should work like a helicopter should better have a large rotor like a helicopter, because it takes much less power to spin it (and thus has much lower fuel consumption).