No, there are no lights left illuminated on the dashboard instrument cluster when the car is off. The only lights that remain on are: the alarm light - flashes when armed; parking light if switched on; and interior lights if you switched them on or have the door(s) or trunk/hood open.
I presume you are getting a jump to start the car and it sounds like it runs fine from there? Then you park overnight and it is dead in the morning? Sounds to me like you have an ongoing electrical drain, which you need to locate. Could be a light left on, something miswired, etc. The cigarette lighter socket stays powered w/the key off, so any accessory there could be a drain as well.
I'm just guessing, but I think your battery is fine (but may be on the way out if this deep cycling continues)... an alternator-based system requires a battery to keep the engine running; if yours was truly dead the car would not run when disconnected from the "jump".
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2001 Lapis/Black/Black, PSM, Rear Speaker Kit, Optima...
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