You can, via spore printing, however you will have to worry about senescence.
Senescence:
the condition or process of deterioration with age.
loss of a cell's power of division and growth.
In layman's terms: the mycelium culture can only be propogated from grow to grow for so long, depending on the culture (which is generally unknown and variable), before it weakens and becomes unusable. I've heard estimates of about 20 times, but I have no evidence that this number is true. You can defeat this by using master culture slants.
Master Culture Slant: similar to using petri dishes with agar to isolate a culture, you can use master culture slants, stored in your refrigerator to slow the growth of the mycelium you store in it, allowing you to reuse the "master" culture many many times--thus allowing you to continue growing a precious strain for many many years.
Spore Printing:
When the caps spread and begin to drop their spores, carefully cut one off and place it on a piece of tin foil inside a SAB (still air box) or inside a sealed mason jar. Sterility is of course still important during these steps. After the cap has finished dropping it's spores, seal the tin foil and store inside a plastic bag inside your refrigerator. You can then use these spores to create new grows, however you wish.