Sure, you can freeze them, but they won't be as good as fresh ones. But a frozen pawpaw is still better than a rotten one. Sort of like the difference between a freshly ripe banana and a frozen one. They will still taste pretty good. You can freeze them whole or scoop out the good stuff and freeze it in a container.

I usually just scoop or squeeze the pulp out into a container(remove the seeds) and freeze it. Works OK for later use in cooking.