B737-800 pilot reported a flap indication issue during final approach; resulting in a terrain warning. The aircraft was reportedly on profile; so the crew elected to continue the landing.

Narrative

At 1;500 feet AGL on final to ZZZ XXR I called for 'Landing checklist - below the line'. At 'Flaps' it was observed that the flap handle was at 30 but the flap position indicator displayed flaps 1.We elected to increase airspeed to the white caret on the airspeed indicator and proceed with the landing. We did not set the Ground Proximity Flap Inhibit switch to 'Flap Inhibit' which resulted in receiving the 'Too Low - Terrain' aural warning. The aircraft was on profile so we ignored the warning and landed normally; with a slightly longer than normal rollout.After touchdown the 'PSEU' (Proximity Switch Electronic Unit) light illuminated and the flaps moved to 30. We were able to raise the flaps after clearing the runway; subsequently the 'PSEU' light extinguished.Cause - We elected to land in lieu of going around in an aircraft with unknown possible flight control issues. Aircraft was on profile on a clear night with light winds and a long runway. The alternative was to take the aircraft back into the air to perform the QRH procedure and attempt an alternate flap extension procedure.Suggestions - This would be a useful training scenario.