I wonder if there are ways to safely run metal halide, high pressure sodium, and mercury vapor lamps off a battery operated DC circuit. If so, I would be interested in making battery operated DC powered metal halide, mercury vapor, and high pressure sodium fixtures. In this context, I am excluding metal halide lamps used for car headlights.

You can run 35W CMH lamps on a DC ballasts for MH lamps for car HID headlights, which are also 35W, but the ballast may overload the electrodes of the lamp, during run-up, and damage the lamp if it isn't suitable for instant hot restrike.

If the lamp isn't designed for DC, running them on a DC supply would likely cause uneven wear on the electrodes, causing early failure. The only realistic way is to use an inverter and a normal ballast, but a high power lamp would quickly drain a car battery, so you'd need a much bigger one or a DC power supply.

I think he asked about a DC driven ballasts, that allowing to run HID lamps off battery (But the lamps runs on AC).

I think he asked about a DC driven ballasts, that allowing to run HID lamps off battery (But the lamps runs on AC).

Indeed, but if the lamp isn't designed for DC it will cause uneven electrode wear. A DC-driven inverter feeding a standard ballast is the most practical way to go, but if you are running anything but the smallest lamps, you will require a very large battery. Lead-acid batteries as used in cars aren't designed to be deep-discharged anyway. Batteries that will tolerate this exist, as for example those used in mobility scooters and electric wheelchairs, but they are considerably more expensive.

They make and sell HID flashlights for this very reason...

They make and sell HID flashlights for this very reason...

I didn’t even realize those were a thing...
Maybe I should pick one up... 

I have a couple of them I bought a number of years ago . They're difficult to find now as LED has wiped them out also !!
 https://www.lighting-gallery.net/gallery/displayimage.php?album=5359&pos=3&pid=171864
 https://www.lighting-gallery.net/gallery/displayimage.php?album=5359&pos=2&pid=172001

The main reason for virtually no existence of any DC powered commercially made gear for standard HID is, the HIDs are devices performing well virtually only when operated for many hours for each start. Plus they are rather high power devices (mainly for that use pattern), so consume rather high amount of energy, so would need rather large battery for that.
Such high power, long hours independent systems usually use a mechanical energy source (an engine of some sort), where it is theeasiest to use AC power distribution (e.g. larger ships use dedicated engine for electrical power generation, portable light masts have their gensets,...), so use rather standard AC equipment (because it makes the system rather cheap and versatile).
Or it is just more convenient to use there an inverter, as it is used for many more things than just the lighting (smaller work vessels; using the main engines only to supply electrical power)

There are few niche applications with DC power system, but these use dedicated gear for the task (car headlight, HID flash lights), because these usually have special requirements for the HID as well (typically nearly instant full brightness,...) and so may get away with worse performance in other aspects (lamp life in the few 100's till 2k hours instead of standard 16k hours, 20..30kV ignition voltage, need for boosted power after startup,...), but which make their ballasts not compatible with standard lamps.

So there is no need for such thing in a normal market.

Of course, DC powered ballast for such lamps is possibly to design (after all an auto HID ballast is using a topology perfectly suitable for all pulse started HID, it just includes some special extra features needed for the auto application).