Lanzante, the car's maker that is renowned for its work on customer McLaren F1s and F1 GTRs, clocked a 6min 43sec laptime at the Green Hell with its machine, which is 9sec quicker than the latest record holder, the Lamborghini Huracán Performante.
To emphasise its road worthiness, Lanzante said the car then drove off of the circuit and home to the UK.
The P1 LM is claimed to be the quickest, rarest and last of the P1 variants. Just five production models are being made.
Lanzante says the biggest change from the P1 GTR is in the engine hardware, which now produces additional boost and hybrid power to keep the full 986bhp from the P1 GTR. It also gets gold-plated heat shielding in the engine bay, and the charge coolers have been tweaked to be more efficient at higher temperatures.
The P1 LM is 60kg lighter than the P1 GTR thanks to a number of weight-saving measures. The exhaust headers and catalytic convertor pipes are made from Inconel, saving 4.5kg, and race parts like the air jack system have been removed and lightweight seats from the F1 GTR are used.
Other alterations include fully exposed carbonfibre for the roof and additional panels, as well as improved aerodynamics from a modified rear wing, a front splitter and dive planes that, Lanzante claims, increase downforce over the P1 GTR by 40%.
Inside, the car gets exposed carbonfibre for the dashboard, instrument cowl, seat backs, roof, centre console and even floor mats. Air-con comes as standard.
Along with one grey experimental prototype, four production models will be produced in orange and one will be grey.
McLaren test driver Kenny Bräck has been part of the team that's developed the car's final settings.
Lanzante decided to undertake the conversion work following demand from customers who have bought the track-only version of the P1 GTR from the factory.
More than 40 £1.98 million P1 GTRs were sold, with the cars being offered to existing P1 road car owners only. The track car has a kerb weight of 1440kg, giving it around 685bhp per tonne.
Neither the McLaren factory nor its MSO division has been directly involved but McLaren is believed to have given the project its blessing. "I wouldn't do anything to damage our relationship with McLaren, which we've built up over 20 years through our work on the F1," said Lanzante boss Dean Lanzante.