Prosody Transfer (PT) is a technique that aims to use the prosody from a source audio as a reference while synthesising speech. Fine-grained PT aims at capturing prosodic aspects like rhythm, emphasis, melody, duration, and loudness, from a source audio at a very granular level and transferring them when synthesising speech in a different target speaker's voice. Current approaches for fine-grained PT suffer from source speaker leakage, where the synthesised speech has the voice identity of the source speaker as opposed to the target speaker. In order to mitigate this issue, they compromise on the quality of PT. In this paper, we propose CopyCat, a novel, many-to-many PT system that is robust to source speaker leakage, without using parallel data. We achieve this through a novel reference encoder architecture capable of capturing temporal prosodic representations which are robust to source speaker leakage. We compare CopyCat against a state-of-the-art fine-grained PT model through various subjective evaluations, where we show a relative improvement of $47\%$ in the quality of prosody transfer and $14\%$ in preserving the target speaker identity, while still maintaining the same naturalness.