Can penguins climb

By the basic definition - no, but when they approach land from the water, they can build up a lot of speed and launch themselves out of the water to even higher altitudes. They need to do this partly to get up on the ice and get away from the edge of the water where their predator (the leopard seal) hunts.

However, from a standing position on land it is impossible for a penguin to jump 6 feet straight up.