PA-28 pilot reported a runway excursion during a night training flight with an instructor resulting in nose gear strut damage but was able to taxi to the ramp.

Narrative

I was enrolled in an Instrument Pilot training program. This course was based at ZZZ.I have a private pilot certificate from the UK CAA which has been recognized with equivalency under ICAO; and I was learning to fly towards an Instrument Rating under full time instruction from a CFI.On Day 0 around XA:45. I was engaged in a VFR Night approach into ZZZ Runway XX whilst under instruction from a CFII. We had completed supported and instructed Instrument Approaches across the general area. The IFR Plan was canceled as there are no Instrument approaches approved for ZZZ at night. The visibility was clear and there was no cloud in the area. The Runway was illuminated and visible on approach. Runway XX has 3;000ft of usable Runway which was dry. The landing speed was relatively high around 68-72 IAS; leading a conscious effort to stop timeously. As the Aircraft;a Piper PA28 181; touched down on the center line; braking was applied in a fashion I am usually accustomed to and was trained in. There was nothing to my knowledge unusual about the braking application; however; after only a second or two; the left wheel locked directing the aircraft to the left. The aircraft made a brief excursion onto the grass to the left of Runway XX before correcting back onto the Runway with a right hand input to the rudder by me. Power was at idle. The aircraft then stabilized on the Runway again.The aircraft taxied back to the FBO under its own power. No damage was noted to any Runway lighting but the aircraft has sustained nose gear strut deformation and other associated damage.I am unable to determine the reason for the aircraft's deviation to the left sharply; there always remains the possibility of asymmetric braking force (this is most likely given the locking of the wheel) combined with hard corrective rudder input; but it could also have been a dual input from the CFI on the brakes; leading to abnormal braking force as this isn't the usual feel of the aircraft for me. I am aware of the importance of a stabilized approach touching down at the appropriate POH airspeed and relevant wind/weather factors. Braking should be applied in a safe; progressive; and controlled manner; in a way that force is equal across the aircraft; not in a hurried and sharp way. If the incident was a dual input situation which remains a possibility; there needs to be clearer communication from the CFI as to who has control of the aircraft's control systems at all times; and whenever I fly with a CFI again; this will be my utmost priority as part of the briefing before flight and before maneuvers/approaches and landings.