In a large mixing bowl, beat the softened cream cheese until smooth and fluffy.
Add the sweetened condensed milk and vanilla extract, mixing until combined.
In another bowl, whisk together the instant pudding mix and cold milk until thick, about 2 minutes.
Fold the pudding mixture into the cream cheese mixture until smooth and creamy.
Gently fold in the whipped topping, being careful not to deflate the mixture.
Line the bottom of a 9x13 inch dish with one layer of Chessmen cookies.
Add an even layer of banana slices over the cookies.
Pour half of the pudding mixture over the bananas, spreading evenly with a spatula.
Repeat the layering: another round of cookies, bananas, and the remaining pudding mixture.
Top with the final layer of Chessmen cookies. Press them gently so they set into the pudding slightly.
Cover and refrigerate for at least 4 hours, or overnight for best results.
Before serving, chill until firm and slice into squares. The top cookies will soften slightly, giving the pudding a lovely cake-like texture.
