A student pilot reported a NMAC in the VFR pattern at a non-towered airport.

Narrative

Climbing off the departure end of runway XX at ZZZ student had just turned the landing light off and set the prop (happens at 400AGL) when student noticed traffic off his right side at roughly the same altitude before attempting to make the turn to right crosswind (roughly 500 ft AGL). Student pulls up and turns right as an evasive maneuver; missing the traffic by approximately 200 ft vertically. Student notes that the traffic was a low-wing that was white with a brightly colored stripe; but did not know the type or catch a tail number. It 'kind of looked like a piper' said the student and he remembers seeing a tail number but everything happened so fast that he doesn't recall any of it. Student continued to the downwind; base; and final legs; landing and parking the craft. Instructor on the ground was in contact with student on handheld radio; heard him make a traffic declaration and saw the student make the evasive turn. Instructor also saw the offending traffic pass by east to west under the student. Local tower was contacted but had little information due to the low altitude of the incident.