Based on my experience having designed lots of digital logic over the past couple decades, you can make faster chips. But you can't do it by just running existing chips faster than they were designed to work. You have to come up with faster transistors and circuit designs.
You can't do it by just running existing chips faster
The heat produced by a modern CMOS based processor is proportional to its operating frequency. Running it at say 2X its rated speed would produce double the heat. All objects have thermal mass, so they take time to heat up. Double the heat would be fine for a short duration. So in that sense it could be done.
You can even extend the time by increasing the thermal mass. For example by making an insulating oxide layer on the back of the silicon and then attaching a piece of copper to soak up the heat.
The real problem is that it takes a certain amount of time for transistors to switch on/off, and for signals to propagate. It is the register to register timing that ultimately determines this frequency. If you take a processor made to run at some maximum frequency, and then run it at double, you will will probably get lots of corrupted calculations. Even one bad calculation is often enough to destroy the operation of a computer program.
For example, consider a multiply circuit fed by a pair of input registers and ends at an output register. Lets say that it takes 1 nanosecond for the inputs to propagate through all the logic to the output register. Then you can at maximum clock the circuit once per nanosecond (a 1GHz max operating frequency). If you try and clock it at say 2GHz the output register will just have garbage in it.
Most chips have some timing margin built in to the advertised operating frequency, but at most you are going to get around 20% more computing power by over-clocking, nothing too game changing.
The only way to make a CMOS processor run faster is to get faster transistors or circuit designs.
Use super-conducting CMOS processors
TRW corporation once had a superconducting processor project that aimed to do just that. MOS transistor gates are essentially capacitors with some series resistance. The outputs of MOS transistors essentially look like resistors when they are switched on. The amount of time it takes for the transistors to switch each-other on is in large part controlled by the product of the gate capacitance (C) and driving resistance (R).
If you can make R = 0 by using super conductors then the only thing limiting you is that charges take some time to move, and the fact that electromagnetic fields are limited to propagating at the speed of light. That method can theoretically make very fast chips. But it requires super-conductors, which in turn requires cold temperatures.
If you only need it for a short time, then you could have an insulating case (like a thermos, or aerogel material), about the size of a small drinking flask, filled with liquid helium or liquid nitrogen as well as your small computer. As long as the processor was off, you could carry the container for days. At any time you could switch on the device and run it until your coolant heated up too much for your super-conductors to work.
The user interface doesn't need to be any more complicated than a cheap smart phone and is on the outside of the insulation barrier and connects to the processor inside the cold area.
Quantum computers are large, but they don't have to be...
Most quantum computers are very large. For example the D-Wave quantum computers are the size of a room and have a very large refrigeration system. The fact is, these systems don't need to be so large. They are designed to be the size of a room because they are experiments, and as such its useful if two or three scientists can walk inside and debug stuff. Also most of the components are off the shelf rack-mount equipment, because the quantum-computer company wants to work on the quantum part of the project rather than distracting themselves by designing lots of custom peripheral components.
The actual quantum processor in these systems is the size of a postage stamp. And if you skip the refrigeration system and just use some temporary portable coolant, it could fit in your pocket.
Use a remote computer.
You can of course also have a computer of any size somewhere else and simply connect to it. The processing is done remotely and the answer is sent back to the user almost instantly. This is the essence of "cloud computing". The only downside to this approach is that someone can jam your communications. Or disable your data center without you knowing it, and then whey you try to use your device you are out of luck.
Use more efficient computer architectures
DARPA has program called SYNAPSE that contracted IBM to develop a chip called TrueNorth which was a neuron like computer chip. This chip was significantly more energy efficient than conventional CPUs. It could perform the equivalent of hundreds of millions of multiply accumulate operations per second on as little as 60mW of power. They did a demo chip with 256 neurons, and then scaled it up to 1 million neurons with the next version. There is no reason they couldn't go up to a few billion neurons and totally outperform any other conventional computer out there in terms of energy efficiency.