The deer (사슴, saseum) is read through grace and gentle, lasting fortune — like the turtle, it belongs to the traditional 십장생 group of longevity symbols in Korean art, so a deer dream carries some of that same quiet, long-term favorability rather than sudden excitement.
Korean folk religion also pictures 산신령 (mountain spirits) as often accompanied by, or riding, a tiger or a deer — placing the deer in genuinely elevated spiritual company, not just a pretty forest animal. A deer appearing calmly in a dream can carry a faint echo of this guardian-spirit association, especially in a mountain setting.
A white deer specifically is read as especially auspicious, echoing similar white-animal traditions elsewhere in this category (white snake, white rat) where the color intensifies an already-positive reading. A deer approaching you peacefully, rather than fleeing, is read as a gentle blessing arriving without any real effort required on your part.
A deer running away is read more as a fleeting chance at grace or recognition — something good that requires a calm, gentle approach to actually catch, since deer in general symbolism rarely reward force or hurry.