In the N.T. there are two words for wilderness, one is used thirty three times...
G2048
ἔρημος
erēmos
er'-ay-mos
Of uncertain affinity; lonesome, that is, (by implication) waste (usually as a noun, G5561 being implied): - desert, desolate, solitary, wilderness.
The other is used twice and is derived from the first...
G2047
ἐρημία
erēmia
er-ay-mee'-ah
From G2048; solitude (concretely): - desert, wilderness.
In both cases, solitude and being alone are the primary meaning, with the implication of a desert. If we think about this, a spiritual wilderness would be a place of being alone in a dry (spiritually) place.