You also may wish to choose "check disk for errors" in the USB system's boot menu to check the install media, maybe your installer got corrupted. But it's also possible that your SSD
is the problem.
I'd try formatting the drive manually, instead of through the installer. It still might not work, but the errors might be more definitive.
So, boot off the USB drive, then run partitioning and formatting software. The friendliest GUI software is probably GParted (I think it might be called "Partition Editor" under "Administration" or "System Tools"). Personally, I prefer to use terminal tools, like "fdisk" for partitioning and "mkfs.ext4" for filesystem creation. It's up to you.
Start up the partition editor and setup the necessary partition(s). If you're only going to be booting Ubuntu, I'd go with a 4-6GB system partition, and the rest as a user partition, mounted at "/home". You could also throw in a swap partition of a gig or two, but I prefer running without swap.
If the partitioning/formatting succeeds, you can proceed with the installation. Just point the installer to the partitions you created, instead of letting it do it's own thing.