Pro Mini 5V, input current on RX pin inhibits reset?

I've got a Pro Mini 5V powered by a 9V battery. Power to the device is controlled by a Pololu latching power switch (Pololu Pushbutton Power Switch SV (old version)). I communicate with the device via a 2.5mm stereo plug which I wired up to an FTDI USB 5V TTL serial cable (so RX, TX, and GND only). Communication only occurs on power up as pins 0 and 1 are multiplexed to a motor driver. On startup, if there's serial data, it does some stuff, then powers off using the switch. If not, it does some other stuff.

When the switch is off or the battery is disconnected, plugging in the serial cable causes the power LED to light dimly and the activity LED to blink rapidly, also dimly. I guess I understand the activity LED as that's connected to a serial pin (right?), but I'm not sure about the power LED. Upon first connection, I can communicate with the Arduino fine, but as soon as it tries to power off, it can't seem to. Well, it appears to be powered off, and there's definitely no current coming from the battery, but when the serial cable is connected, it seems to be stuck in some low-power state and I have to disconnect and reconnect the serial cable or reset the Arduino to be able to communicate with it again.

What's going on? I should note that this also happens without using the power switch.

You mustn't provide logic HIGH to signal pins for a powered-down logic-chip - if you do the protection diode on the relevant input(s)
will attempt to carry enough current to power the entire chip/board. Result is either a burnt-out protection diode, damage to whatever
was generating the signal, and possibly latch-up of the chip, or you are lucky and the effect is only temperary and it still works afterwards.

The problem can be mitigated by limiting the current (say with 10k resistors), but really this situation is best avoided - don't power down
the Pro Mini, put it in sleep mode instead.