Etsy seller AmeliaMakesArt makes a mini-likeness of your dog out of wool:
I am in love with making these little needle felted dogs and would love to make one for you! I can make a dog of the breed you love or I can make a portrait of your dog from photos you would email me.
Each little pooch is sculpted out of Icelandic and Merino wool. The technique is needle felting and uses a special barbed needle tool to create the form. Each dog is one of a kind and sculpted in fiber like one would sculpt clay. There is no sewing, stuffing or pattern of any kind.
Link - Thanks bethany!