SharePoint Online Access Rights on Shared Folders.

aero2466This is Surely Possible you will have to create custom permission with Contribute no delete option...

Creating New permission Levels

Go to the gear icon on the Sharepoint > Site Permissions > Advance Permissions Settings > Permission Level > Add A permission Level Or open Contribute and at the bottom, it will say copy permission Level do so and name it > Name the new Permission and uncheck the Delete Item and Delete Version > Click Create

Now Go back to the Folder/Library > Manage Access > Advance > Stop inheriting Permission and add the members you want > While adding select Show options > in Select a permission level = chose the recently created permission whatever you name it could contribute no Delete > Click Share

You Are done with this setting let me know if you have any issues i have done the same and it is working fine for me...