In a large container with a fitted lid, combine the vodka, Pimm's, lemon and lime juices, honey, hot sauce and 1 scant teaspoon pepper. Cover and shake to blend. Refrigerate at least 1 hour.
Fill a pitcher with ice and the lemon and lime slices. Shake the chilled vodka mixture and pour into the pitcher. Using a long wooden spoon, stir to blend while adding the lemon-lime soda. Taste. You may find you want to add another splash of vodka. Pour into tall glasses.