Pin values when Arduino switched off

Hello..

My project controls power to an Arduino with a tilt switch, timer and an N-channel Mosfet as a switch.
When the power is off the Arduino is floating with no connection to ground. (Arduino ground is switched through the Mosfet).

The arduino itself is configured to control a different Mosfet to fire a solenoid through an output pin with a 10k pull down resistor. (Pulled to external ground not to the switched ground).

Everything works as expected when the arduino is on. However, when the arduino power is cut by the timer, and the arduino is "floating", the output pin seems no longer to pulled to ground by the 10k resitor and it goes high.

Is this avoidable with the configuration I have? If not - I presume the only solution is to switch the Arduino with a P channel Mosfet on the positive and have it permanently connected to Ground ?

Any advice would be very much appreciated...

When Arduino is unpowered, the pins are in Hi impedance state, like an open circuit, so no pulldown (or up). Can you post a wiring diagram?

outsider:
When Arduino is unpowered, the pins are in Hi impedance state, like an open circuit, so no pulldown (or up).

No. When any electronic chip is unpowered it looks like a diode to ground and to the power rail. It is not high impedance.

sjdevlin:
The arduino itself is configured to control a different Mosfet to fire a solenoid through an output pin with a 10k pull down resistor. (Pulled to external ground not to the switched ground).

You should pull down to Arduino ground only - otherwise you are pushing current through the protection
diodes on the chip (generally not a great idea).

Everything works as expected when the arduino is on. However, when the arduino power is cut by the timer, and the arduino is "floating", the output pin seems no longer to pulled to ground by the 10k resitor and it goes high.

The Arduino pin is at Vcc when the Arduino isn't powered, so all its pins will be at best 0.5V below Vcc, hence
your problem.

High side switching would have been less problematic.