Where did you see 「いいえ、私のやつではないです。」? Almost no one would say that in real life because 私のやつ is collocationally constrained. やつ sounds fairly slangy and mannish, so it does not sound natural to combine it with 私. One will, however, occasionally hear オレのやつ or even ボクのやつ, both meaning "mine" when both the speaker and listener know exactly what item they are talking about.
Native speakers will just use 私の without a noun following it since 私の, in this particular context, already means "mine" instead of "my".
「私のではありません。」 = "It is not mine." More literally, "It is not my belonging."
In the sentence above, the 私の part can be replaced with 私のもの but just using 私の would be a more natural choice for us native speakers. もの may be added when wanting to emphasize that the item belongs to no one but to you yourself.