No. If you make a derivative work that is based on a public domain work, you can only claim copyright in the derivative work, not in the original public domain work. Creating a derivative work that incorporates elements of a public domain work does not “resurrect” the copyright that has entered the public domain. Once a work loses copyright protection and enters the public domain, it stays there.
Copyright may be claimed in the modifications to the work that are made in a derivative work, but not in the public domain work. It will be necessary to exclude the public domain material from your claim if you seek to register a claim in the derivative work.
If the two derivative works are similar to each other but were independently created – that is to say, if no copying of one derivative work by the creator of the other derivative work occurred – then two different people could be owners of copyrights in the same derivative work. The probability of this occurring without even subconscious, unintentional copying is not necessarily great, but it is within the realm of possibility.