To make the caramel, combine the corn syrup, sugar and water in a small saucepan and cook over medium-high heat without stirring until it turns a deep amber color, about 10 to 12 minutes.
Decrease the heat to medium and add the cream. Continue cooking until the mixture smooths out, as it may seize up when the cream is added. Add the butter, vanilla and salt and stir until incorporated.
Let cool to room temperature on the counter before using.
Place a 9x5x2-inch loaf pan into the freezer about an hour before making the ice cream.
In a food processer fitted with the blade attachment, process the cream cheese, vanilla and salt until smooth, about 30 to 60 seconds depending on your food processer. Add the heavy cream and process until stiff peaks form.
Add the pumpkin, sweetened condensed milk and cinnamon and process just to combine.
Transfer the ice cream to the cold pan and smooth the top. The pan will be very full. If it is too full for your liking, pour some of the ice cream base into a second smaller container.
Drizzle about 1/2 cup of the caramel over the ice cream base. Use a fork to swirl it throughout the ice cream. Then dollop a 1/4 cup over the top and swirl it across the surface decoratively - this makes for the prettiest of ice creams.
Cover in plastic wrap and freeze for at least 8 hours. Serve scoops with an extra drizzle of caramel on top. The ice cream will keep in the freezer tightly wrapped in plastic wrap for about two weeks and the caramel will keep in an airtight container in the refrigerator for up to a month.