Pilot reported an NMAC on final approach and continued to a normal landing.

Narrative

Returning to our home airport after dark; we flew a straight in visual approach to runway 29. We began monitoring the local UNICOM about 9 miles away; and made position reports then; as well as at 6; 3; and 1.5 miles from the airport. Additionally; we inquired to the FBO about the surface winds; as the local AWOS was out of service. There was no other known traffic in the pattern; until Aircraft Y reported being 'on the RNAV' to runway 29. It was at this time we made the 1.5 mile final radio call. At roughly 200 feet AGL; loud jet whine was heard and we observed Aircraft Y turning and climbing away from a position slightly below and behind our aircraft. We continued to a normal landing; while Aircraft Y made a right traffic pattern before landing. During radio discussion after the incident; the Aircraft Y pilot stated they had been cleared for the RNAV by (presumably Denver) approach; and had seen us on TCAS but not been given an RA. They further stated that they were a single pilot operator and had understood us to be crossing the final instead of landing.