Is it possible for Hydrogen to lose its electron?
Hydrogen can lose an electron meaning it can be in the +1 oxidation state. However, just like any other cation or anion it never occurs free in condensed matter, it always is in contact with solvent and/or anions. Moreover, because of extremely small size of proton, it is an extremely powerful Lewis acid. Consequently, in common conditions proton would react with first electron pair it comes in contact with, up to and including inert gas electron pairs and covalent bond pairs. 

On the other hand, hydrogen ions are quite easy to generate in electric discharge and/or under extreme heating. In fact, producing and confinement of super-hot plasma, consisting of hydrogen ions and electrons, is an area of active research for several decades and, well, producing and confinement of relatively cold plasma is not a problem. Confinement of several billions Kelvin hot plasma, however, is still a problem.