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yes I'm afraid so
if the engine is run low / out of oil then the oil pump will be damaged and pick up surfaces so create scoring damage on the pump rotors
however before replacing the pump
I'd 1st suggest measuring the actual oil pressure with separate mechanical gauge - this can be plumbed into the engine via the current pressure switch port - if you use a Tee piece you can keep the pressure switch plumbed in as well and so see if the low pressure light corresponds to a low pressure reading.
As its intermittent and you have no rumbling or rattling noises from the engine its likely to be an electrical fault so 1st check the connection to the pressure switch is bright and clean and check back along the switch wiring for any signs of damage to the insulation
As if the wire is grounded out at any point then this will also turn on the light.
if this is all OK and the oil pressure is at least 1.3Bar ( 20psi) at hot idle then replace the switch
If you do have a low pressure reading and to turn on the light it will be very low about 5psi then replace the oil pump and ensure that the pump pickup in the sump hasn't come loose or is blocked
providing you don't open another question then yes this is all included in the original price
one other thing to consider is running low on oil can results in other engine damage I'm afraid as bearings can be damaged as they're meant to run on a film of oil, if there's no oil to support the bearing you increase the clearances and can score the crank shaft
if the engine rattles on a cold start then this is a sign that the bearings are worn
I have a mechanical engineering degree ( Bachelors) and qualified in 1995 and have been working in the automotive industry ever since
I design, develop and calibrate engines of all kinds on a daily basis have have worked for 12 years in Formula 1 designing engine and car parts