Permanent connections with Arduino Due
What it sounds like you want is an Arduino Due footprint prototyping shield.

Arduino make one, although it is out of stock, but it gives you an idea of what you are looking for. There are plenty of other designs out there for purchase of varying degrees of complexity.

You solder the pins in place and mount it on top of your Due just like any shield, and solder your wires and components in where you want them.

An alternative option, of course, is to step away from Arduino and investigate some of the many other chips from other manufacturers out there that are considerably more powerful than the 8-bit ATMegas used on the low-end Arduinos, and yet are still available in a DIP footprint - chips like the PIC32MX1xx/2xx series from Microchip - 28 pin DIP (just like the 328P) yet 50MHz, 32-bit power.  Yes, it will be a bit of a learning curve, but there are helpful options out there that can assist you, like the chipKIT system including the handy DP32 board from Digilent.