It's not a first-person narrative. From the Wikipedia-JA article (
私小説 - Wikipedia) on 私小説:
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その名称から、「私」と一人称で語られるものとする誤 解があるが、三人称のものも多い。
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(because of the name, "私" is often read mistakenly as "first-person," but there are many third-person 私小説, too")