NEW YORK, Jan 9 - The dollar fell steeply, with the euro surging to new record highs in the wake of a surprisingly weak December U.S. payrolls report on Friday. The report showed a paltry increase of 1,000 nonfarm payroll jobs and a jobless rate falling to 5.7 percent, compared with expectations for a rise of 130,000 jobs and a jobless rate of 5.9 percent. Early morning in New York, the euro shot higher in swift reaction to the U.S. report, climbing more than one cent to new session highs around $1.2852. The dollar weakened broadly, with the pound surging above $1.8400 to new multi-year highs. The dollar slipped to just below 1.2200 Swiss francs, a seven-year low.//