If the controlling software allows it then yes.
There's at least one drone available where autonomous flight is possible ; the Vision 2 FAQ says that the drone will continue its "mission" even if you lose the signal to it, although the overall length of the flight path is limited to 5km and the drone will not fly close to airports.
Other drones stop and hover or land if they lose the connection to the controller.
Whether doing this is actually a good idea (or legal in
whichever jurisdiction you want to operate in) or not is a whole other question.
aviation.stackexchange.com/users/62/Pondlife
In general, unmanned flight control systems operate in one of two main modes: manual control and program .
(There are other modes but these are typically specific situations, such as launch/landing sequences, loss of link (aka "return home"), and any other design-specific operating modes as required (see below).
To answer your question, if link is lost while the aircraft is in manual operation, a good flight system will automatically go into "return home" mode, where it flies to a pre-programmed GPS coordinate and altitude.
Some software will allow you to customize the sequence, such as whether or not to spiral up to altitude before heading to that coordinate.
Other software will backtrace the route it took.
Aircraft following a pre-programmed route typically change to manual mode upon completion, hoping there's a good ground link at the other end.
In military operations, this is known as a "hand-off" where one ground station takes control from another.
One Shadow flight I was running lost link immediately after launch.
After completing its autolaunch sequence, it went to return home and circled for hours until it ran out of fuel, at which point the engine stalled and the generator (alternator) hiccuped in transition to battery power, causing the flight processor to reboot, at which point we re-acquired link, though too late to do anything about it.
The UAV had already noticed its engine wasn't spinning and initiated its ditch sequence, rolling over and popping the chute.
aviation.stackexchange.com/users/6919/Erich
From a technical standpoint, as other answers pointed out, it is technically feasible.
For instance, universities have attempted a 180 km flight over the Mediterranean sea .
However, there are also legal issues involved when flying a drone beyond the range of the remote control device.
For instance, in France , even when using an autopilot to control a drone, a safety pilot must stand ready to take control of the drone in case anything happens.
The Chicago Convention (1944), article 8 states that No aircraft capable of being flown without a pilot shall be flown without a pilot over the territory of a contracting State without special authorization by that State and in accordance with the terms of such authorization.
Each contracting State undertakes to insure that the flight of such aircraft without a pilot in regions open to civil aircraft shall be so controlled as to obviate danger to civil aircraft.
In many countries, these special authorizations can take a few months (a little more than three months in the case of the 180 km flight over the Mediterranean sea) to obtain, so hobbyists usually don't bother.
aviation.stackexchange.com/users/3775/usernumber
Yes, most of the sophisticated consumer drones allow you to pre-program a track into them and can fly totally autonomously.
Also, most of even the less expensive drones (FN1) are designed to return back to their starting GPS coordinates if they lose their control signal.
Fn1
: Less expensive still means around USD 500 or so.
Sophisticated drones are USD 1000 and up.
aviation.stackexchange.com/users/2407/RoboKaren
From a technical perspective, there's nothing preventing it.
It wouldn't even be very difficult to write the code to do it if you had a GPS receiver on board, especially if you also had accelerometers to interpolate position between GPS fixes.
For that matter, remote control via a cellular network wouldn't be particularly complicated from a technical implementation perspective (and this would also aid in navigation if you triangulate position from the towers.)
Where the problem comes in is legal issues.
Specifically, in the U.S., at least, operating such an unmanned aircraft outside the line-of-sight of the operate is illegal unless you have a special license from the FAA, which is generally only granted to federal government agencies or law enforcement/emergency/rescue organizations or to aircraft R&D organizations.
According to the FAA : Recreational use of airspace by model aircraft is covered by FAA Advisory Circular 91-57 (PDF) , which generally limits operations for hobby and recreation to below 400 feet, away from airports and air traffic, and within sight of the operator.
aviation.stackexchange.com/users/755/reirab
Federico was right on in his comment in that this is a weak 'answer' and yes, I know better than that.
While not answering the body, this question does answer the one in the title.
See Man detained outside White House for trying to fly drone , as well as related stories of that day, for an example on what not to do with your UAV... As a synopsis,
if I remember the details correctly, a US Federal employee was at a friends house in Washington D.C. and they were playing with the friend's new quad-copter (some might call it a drone).
There was much drinking and celebration and the man was flying the quad while quite drunk.
He flew it towards the White House and it disappeared.
It landed itself on the White House lawn.
So, yes, a 'drone' can fly beyond radio range.
aviation.stackexchange.com/users/2171/CGCampbell
Depends on the programming of the drone - mostly if it contains code for autonomous operations.
Furthermore, it can have a "drone call home" subroutine which would cause it to become "autonomous"
(it isn't fully autonomous, but it can fly on it own in a straight line).
aviation.stackexchange.com/users/14857/mzdv
In a hobby drone, is it possible to set up the autopilot so that the drone travels to a distance far beyond the range of the remote control device and then comes back to the starting point? When I see hobbyists flying drones , it seems the drone always hovers within the boundaries of the field.  Is it possible for a drone to reach a friend's house on the other side of town and then come back to the start point?
Can a drone travel beyond the range of its remote control signal?
aviation.stackexchange.com
<autopilot><unmanned-aerial-vehicle>
aviation.stackexchange.com/questions/11260
aviation.stackexchange.com/users/5556/Victor
It is feasible for a drone to continue flying when the signal is lost. It depends on its programming. Good drones have "return home" mode. However, in the US, it is illegal for drones to fly outside the line of sight. It is technically possible for a drone to travel beyond the range of its remote control signal provided that its programming and software allows for this. However, it is illegal in the USA to fly an unmanned aircraft out of sight of the operator unless you have a special license from the FAA. Model aircraft for recreational use may only be operated below 400 feet, within sight of the operator and away from airports and air traffic.
It is technically possible for a drone to travel beyond the range of its remote control signal provided that its programming and software allows for this.  In the USA, it is illegal to fly an unmanned aircraft out of sight of the operator unless you have a special license from the FAA. Model aircraft for recreational use may only be operated below 400 feet, within sight of the operator and away from airports and air traffic.