C172 pilot reported low oil pressure and smoke in the cockpit resulted in an expeditious return to departure airport. Pilot reported oil on exterior of the cowling.

Narrative

On Day 0; Aircraft X departed from Runway XXL at ZZZ and climbed out at Vy. While performing the climb checklist; high oil pressure was observed. The nose of the plane was lowered to 85 knots and we began turning back towards the airport. Shortly after XA:25; smoke filled the cockpit. I reduced the power to idle; immediately pitched for best glide of 65 knots; and informed ZZZ Approach of smoke in the cockpit. I was immediately cleared to land on any surface at the airport. Emergency checklist was performed. Oil pressure was in the bottom of the green range and power was applied as needed to maintain enough altitude to make it to the runway. Any time power was applied; smoke filled the cockpit so I only applied power for short durations. The left window was opened to clear the smoke from the cockpit. Best glide was maintained and a no flap landing performed on Runway XR. I taxied the plane to exit at [Taxiway] 1 and shut the plane down. Oil was observed on the outside of the cowling. There was no fire. Airport Fire and ZZZ Airport Operations met us at Taxiway 1. Airport Operations gave us a ride and escorted flight school mechanics to the taxiway to tow the plane back to the flight school hangar.Prior to departure; a full preflight was completed. Oil quantity observed on dipstick indicated 6 quarts. The engine blanket was on the plane and the engine heater was plugged in overnight on the electrical posts at the flight school. A full run-up was performed with no unusual observations. There were two souls on board and no injuries. Aircraft X is a C172 I rented from the flight school. I was PIC.