It's your inner door seal, well it certainly sounds like it.
The inner door seal is a foam sealant, not rubber like the other seal at the bottom of the door.
You need to undo all the screws on the door card and remove it, unplugging any wiring as you go. There are screws along the bottom of the door card and also in the door grip (black handle) which you'll need to prise apart to see. Once this is off, you'll see a large metal plate that is silver in colour and around the edge you'll spot the foam sealant. Now run your eyes around the edge and look for a place where the sealant has perished, usually it's at the bottom of the plate and is made noticable by dirty water stains around that area.
Once you've located it, undo the bolts holding the plate on but don't remove them, just loosen so you can fit almost a finger between the plate and the door. If you do remove the plate, your run the risk of the window dropping or it not working correctly after refitting everything. Anyway, now get some silicone sealant and reseal the part that has perished or like I did, reseal the whole bottom edge so you don't need to do this again.
Plug in your electrics again and connect the latch mechanism to the inner handle, then refit the door card and allow to dry.
When it next rains, all should be well.