Is it possible to mine copper without using fossil fuels?
Of course. That's how they did it in the very old days when copper was the main metal. It's just very labor intensive.
Sure, if you have lots of people with shovels and lots of people with buckets to carry the ore and lots of wood to burn to generate the heat required to roast the ore, etc etc. Primitive people were able to smelt Cu bearing ores to metal without using fossil fuels (the Bronze Age) but they could only do small batches at a time. These days it would be possible to do this all with electric power but most of our electric power comes from fossil fuels. Doing away completely with fossil fuels will take a long time.
copper mining is no different than any other industrial operation. Primary mining equipment uses diesel fuel (haul trucks and excavators) and/or electricity (shovels and processing equipment--pumps, crushers, mills, conveyor belts, electrowinning). Many newer operations tend to no longer employ smelting, though smelting is still an important part of the global copper balance as a source of sulfuric acid required for heap leaching and general copper processing (copper sulfate). Some very clean and modernized legacy operations (Kennecott in Salt Lake) are great examples of a modern smelter--operated on a continuous basis for high efficiency (meaning CLEAN).
