This is more than likely the portrait Donne described in his will and left to Robert Ker, later 1st Earl of Ancrum (1578–1654):
I give to my honourable and faithful friend Mr Robert Karr of his Majesties Bedchamber that Picture of myne wch is taken in Shaddowes and was made very many yeares before I was of this profession [i.e. a minister].
England’s National Portrait Gallery is trying to raise funds to buy the protrait for the Gallery’s collection. You can read about the appeal here, and donate, very easily even from North America. The National Portrait Gallery must raise £1,652,000 by the end of May.
While I’m on the topic of Donne, I want to point to this nifty .pdf chart you can download and print: John Donne on Maps and the Microcosm. There’s even intelligent commentary. It’s an effective and nicely done exploration of the two motifs, and quite useful in teaching. The broadsheet is a production of the University of Wisconsin’s The History of Cartography Project, which has a series of downloadable broadsheets on “Literary Selections on Cartography.”