You can see the diode pointing forwards on the P-channel, so yeah, swap that around.
Also, I think the 10 kOhm pull-up is a bit weak. It will make the P-channel switch slowly, and if it's weak and/or the load is heavy, it might overheat.
Also, the transistor and base current limiting resistor are not needed. A 500 Ohm or 1 kOhm resistor from the Arduino to the gate of the MOSFET is all you need if you're using an N-channel that switches on the bottom -- that will be a lot more efficient. If you still need a P-channel, then use a 500 Ohm or 1 kOhm resistor into the gate of a small-signal N-channel MOSFET (like the BS170) and a 1-2 kOhm pull-up, instead of the BJT NPN you're using.
So, what I would do, assuming the grounding of the lamp is not interruptible or reversible to power (so, must use P-channel):
500 Ohm resistor from Arduino to gate of BS170. BS170 source to ground, BS170 drain to P-channel gate and 2 kOhm pull-up to 12V. P-channel source to 12V. P-channel drain to light.