Large? I imagine a large fragment would be as cataclysmic to Mars as the original event was to the Earth. No, small fragments. I'm saying that bacteria have likely been seeded on Mars many times during our planets' shared history, and certainly recently with the probes and rovers. It doesn't appear to have taken, or at least, in no place we've looked.
One idea I have heard is that life may have started on Mars during its early and more idyllic phase of existence, and was since transmitted to Earth. But there's no real evidence is support of this over other explanations, as with all panspermia hypotheses.
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