Variant of 'hitori-kiri' meaning completely alone; the small 'っ' is a spelling variation used in speech and writing for emphasis.
彼女は部屋で一人っきりで絵を描いていた。
She was painting alone in her room.
長期の出張で家族がいないと、一人っきりになる時間が増える。
With long business trips and the family away, the time spent completely alone increases.
寂しさを感じたので、一人っきりで散歩に出た。
Feeling lonely, I went for a walk all by myself.