Can an object in space have a high kinetic energy, but at the same time not release heat?

No heat is generated unless work is being done on/by the object.
As an object in space in freefall is undergoing no force on it, no work is done.
Thus the kinetic energy remains kinetic energy, and just... sits there.

Of course, unless the object is at absolute zero temperature, it will be releasing heat. In the form of normal thermal radiation. But I'm quite sure this is not what you refer to.

No object have kinetic energy by itself. The kinetic energy is always in relation to something else.( sometimes called frame of reference )
In every day of life is usually the earth, but can actually be anything. In relation to earth I have no kinetic energy atm. But in relation to the sun, moon or anything that is moving relative the earth, it is quite high.  It is also large enough i relation to the lorry driving outside my house, large enough to kill me. 

If the temperature of the body is at absolute zero (or at least below the vacuum temperature) then yes. On a macroscopic level there will be no transfer of kinetic energy into internal heat energy. To dissipate kinetic energy into undirected, random, internal kinetic energy (=heat) you need forces to act on the body. As there are no in space the kinetic energy will stay constant.

On a quantum mechanical level you will always have fluctuations so that in practive you never reach absolute zero and thereby will always have some (maybe neglible) radiation.

Can an object in space have a high kinetic energy, but at the same time not release heat?

Yes, because the kinetic energy of the object as a whole, its macroscopic kinetic energy with respect to an external frame of reference, has nothing to do with it radiating heat, which is a function of its temperature relative to the temperature of it surroundings. In this case, a temperature above absolute zero.

The temperature of the object is a function of the random microscopic motions of its atoms and molecules, its internal energy, not the kinetic energy of the object as a whole.

Hope this helps.