Are you wanting to do it yourself? Take a drawer out from a cabinet and find a matching section of lino underneath to cut out. Cut it out bigger than the “wood” piece the damage is on. Tape that piece on top of the damaged piece lined up perfectly with the floor pattern. Cut both pieces at the same time along the “wood” pattern in the lino, this way the new piece will be a perfect fit. Glue the new piece in place of the damaged piece and then use seam sealer along the edges.
A heat gun and the back of a spoon will work wonders.
Heat gun and a rolling pin. Be careful and ease into it.