Yes, filling up disk space will affect performance. SSDs tend to chose empty blocks to write in, since they can only modify a block by erasing and re-writing the entire block. Once all of these blocks are partially filled, then every write operation will require an erase and rewrite, meaning the SSD has to read the block into cache, modify the cache with the new data, then erase the block and write the cache. Having a SSD completely filled up all the time will also shorten the lifetime of it.