Combine non-dairy milk, pudding mix and cornstarch in a small saucepan. Whisk together until the dry ingredients are incorporated. Cook over medium heat, stirring constantly, until it begins to thicken and just starts to bubble. Remove from heat and cool for a few minutes.

Mash two of the bananas to a wet pulp. If you don’t want any chunks in the pudding, you can also puree them in a blender.

Transfer the warm pudding to a large bowl. Add the mashed bananas and vanilla. Fold together.

Cover the pudding with plastic wrap making sure to press the plastic to the top of the pudding to prevent a skin from forming. Refrigerate 1-2 hours until completely cool.

Remove chilled pudding from the refrigerator. Add half of the whipped cream to the pudding and fold together.

Using a 9x9 baking dish, begin assembling the banana pudding. Start with a layer of pudding (about 1⁄4 of the total amount of pudding), followed by cookies and then banana slices. Then spread 1⁄2 of the remaining pudding over the banana layer and repeat with cookies, bananas and the remaining pudding. Make sure the top pudding layer completely covers the bananas to help keep them from turning brown.

Refrigerate for 4 hours or up to 24 hours before serving.

Immediately before serving, decorate the top with the remaining whipped cream and some additional cookies or banana slices if desired.