、。
She still hadn't given up hope of finding her dog alive.
Literal
She [topic-は] her own dog [subject-が] alive be-found [called/that-という] hope [object-を], still was-not-discarding.
~という here labels the noun 希望 ('hope') with the content of the embedded clause: 'the hope that her dog would be found alive.' 生きて is the て-form of 生きる ('to live') functioning adverbially — 'alive, in a living state.' ~ないでいる ('to remain without doing X') describes the ongoing state of not having done something — combined with まだ ('still'), it gives 'still not having given up.' 捨てる ('to throw away') applied to hope is a metaphorical extension — to abandon, relinquish.