。
She said she had never been asked to go on a business trip.
Literal
She [topic-は] business-trip do [as directed-ように] asked [experience-ことはない] [quotative-と] said.
A complex multi-layered sentence. At the core: 出張するように頼まれた = 'was asked to go on a business trip,' using ~ように頼む (to ask someone to do X) in the passive. Wrapping that: ~たことはない = 'has never experienced (being asked).' Finally, と言った quotes the whole thing as reported speech. 出張 (business trip) is a common workplace term — 出 (go out) + 張 (extend/post).