like a deer in the headlights

be like a deer (caught) in the headlights AND be like a rabbit (caught) in the headlights (too nervous to move or speak; so frightened or surprised that you cannot move, speak, or think) — застыть, замереть, оцепенеть (от страха, ужаса, изумления, удивления); растеряться и не знать, что делать (от страха, удивления); оцепеневший от ужаса; испуганный, удивленный; застигнутый врасплох

Example 1: Each time they asked him a question he was like a deer caught in the headlights. (CIDI)

look at sth like a deer (caught) in the headlights AND look at sth like a rabbit (caught) in the headlights — смотреть ошеломленно, удивленно, растерянно, тупо, испуганно, затравленно; ~ смотреть как баран на новые ворота

Example 1: When you meet me, don't tell me any Russian jokes about language or I will simply stare back at you like a deer in the headlights. (chaika)
Example 2: I like the one about the "horse marines," jeez that is absolutely *ancient*! If you use it, be ready for the "deer in the headlights" look, or equally slangy but more contemporary, "Say what?" (chaika)

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A deer is crossing the highway and a car is coming. The headlights of the vehicle will cause the deer to "freeze." Too scared to get out of the way. It is a very hazardous situation and deer are known as the most dangerous animal in the United States because of the resulting wrecks.

And there is the figural meaning, or extended meaning. "He had that look, like that of a deer caught in the headlights." (phrases.org.uk)