Q:

Could a cubesat propel itself to Mars?

I wrote the answer below to the question Could a cubesat be propelled to the moon? before realizing that it said Moon and I'd written it for Mars, so I've cloned that question and moved the answer here.

Is it possible with current technologies to propel a cubesat, which is launched from Earth, to the Moon Mars?

A:

I am assuming you mean by propulsion by the CubeSat itself.
Not at the moment! Mostly because of the throughput (thruster lifetime) constraint on small Electric Propulsion (EP) thrusters designed for CubeSats.
Right now the leading CubeSat EP thruster is the BIT-3 (this is the thruster that will be used to go to the moon on my answer to your original question).
Here are the relevant specs:
ISP: 3500
Thrust: 1.4 mN
Thruster life: 20,000 Hours = 2.28 Years
Assuming a 20 Kg 6U CubeSat, here is a non-optimal low thrust trajectory simulation.

This takes 2.36 Years of thrusting time which is higher than the thruster life of 2.28 years. However, we are very close to this being possible. This simulation doesn't account for inserting into a Martian orbit or inserting into an earth escape orbit from a launch orbit. Both of those would further violate the throughput constraint. 
As a last word, many people wrongly assume this would use a lot of propellant. This is false. The above simulation only uses 3.04 Kg of propellant out of a total mass of 20 Kg which is actually small when you think about it. Propellant is not the problem when it comes to EP.

A:

Let's look at some possible examples, building on @ben's answer and @ Knudsen's answer.
We know that the MarCo cubesats were able to navigate from Earth to Mars, with

attitude control via reaction wheels and cold gas thrusters
science data and image collection
communication directly with Earth via a unique pop-up flat high gain antenna
70W of solar power at 1 AU via two deployable solar panels plus battery storage
standard 6U form factor

for more see this answer and links therein.
So let's adopt the MarCo design. They didn't provide their own propulsion, so let's add a propulsion system directly to MarCo's 6U, 14kg initial configuration, and call it 10U and 22 kg. The extra 4U volume is mostly for engines and extra propellant, the extra 8 kg mass budget is for engines and additional solar panels for more electric power, especially out near Mars and a whole bunch more propellant!
Looking for at least apparently existing cubesat electric propulsion systems that you could put in a 3U cubesat today (or soon), the first one that came up in my search is the IFM Nano Thruster for CubeSats. I am sure thee are other options out there, let's just use this as an example. According to that page:
Dynamic thrust range        10 μN to 0.5 mN
Nominal thrust              350 μN
Specific impulse            2,000 to 5000 s
Propellant mass             250 g
Total impulse               more than 5,000 Ns
Power at nominal thrust     35 W incl. neutralizer

Our cubesat will have nearly enough electric power for two engines at 1 AU, since we've expanded the form factor by 4 U and mass budget by 8 kg, let's assume we've found a way to double the size of the solar array to power our new engines. We have now 140 W at 1 AU and ~60 W at 1.5 AU near Mars.
Let's assume our cubesat starts in circular LEO at 400 km with an orbital velocity given by the vis-viva equation:
$$v^2 = \frac{GM_{Earth}}{a}.$$
With $a=(6378+400) \times 1000$ meters and Earth's standard gravitational parameter $GM_{Earth}=$3.986E+14 m^3/s^2, the orbital velocity is about 7700 m/s. 
To achieve Earth escape velocity, and put it in a heliocentric orbit, @MarkAdler's answer tells us that the delta-v necessary for a slow low-thrust spiral outward to escape at very low velocity relative to Earth is equal to the orbital velocity at the start. 
Delta-v from LEO to heliocentric is about 7700 m/s via low-thrust spiral.
Going from 1AU to 1.5 AU we can re-apply the same answer, which also tells us that the delta-v necessary to transfer between two circular orbits is simply the difference in their velocities.
Using the standard gravitational parameter of the Sun $GM_{Sun}=$1.327E+20 m^3/s^2, 1AU ~ 1.5E+11 meters, and 1.0 and 1.5 AU as Earth and Mars orbital distances, we can get the velocity difference to be 29700 m/s minus 24300 m/s or about 5400 m/s.
Delta-v from 1 AU to 1.5 AU heliocentric is about 5400 m/s via low-thrust spiral.
Our two off-the-shelf engines with 250 g propellant tanks each can provide a total impulse of as much as 10,000 Newton seconds. With an average mass of about 20 kg, that only provides a delta-v of 500 m/s, and we're looking for over ten times that even if we've already gotten to heliocentric at 1 AU. That's based on 500 grams of propellant.
Luckily we'd added 8kg to our mass budget, so if we'd added an extra 5 kg of propellant we'd have a total impulse of 100,000 Newton seconds and a delta-v of about 5,000 m/s.
Conclusion:
A back-of-the-envelope calculation starting with a MarCo-like cubesat with demonstrated capability of going from Earth to Mars, augmented from 6U 14 kg to 10U 22 kg with two existing engine designs and another 5 kg of propellant, we can get from a heliocentric orbit at 1 AU to one at 1.5 AU using solar-electric propulsion.
It's a long, slow spiral, many decades or probably a century. You would need even more propellant to do it faster using solar-electric, but even 50% more would cut your transit time to a decade or so based on some simple calculations I did here.
You'll also need an external booster to give you the delta-v from LEO to Earth escape velocity to a heliocentric orbit first.

below: Source: Emily Lakdawalla's Planetary Society blogpost MarCO: CubeSats to Mars! 
Found in this answer.

MARCO SPACECRAFT: Engineer Joel Steinkraus stands with both of the Mars Cube One (MarCO) spacecraft at NASA's Jet Propulsion Laboratory. The one on the left is folded up the way it will be stowed on its rocket; the one on the right has its solar panels fully deployed, along with its high-gain antenna on top.

An alternative, future propulsion system with even higher Isp and therefore needing less propellant mass:

http://neumannspace.com/science/
https://spacenews.com/more-startups-are-pursuing-cubesats-with-electric-thrusters/
Will the Neumann drive start testing aboard the ISS some time in 2018?
Which way will the Neumann drive (on the ISS) point, what will be its maximum possible thrust?

An encouraging video:

