Clearly not, for the Hebrew scriptures detail other animals used sacrificially with regard to various sins. Different animals were used to symbolically "deal with" sin. No animal sacrifices could actually take sins away for "It is impossible for the blood of bulls and goats to take away sins" (Hebrews 10:4). But because blood represents life, the shed and sprinkled blood of various animals was used to teach sinful humans the cost of their sin being dealt with by God. Life had to be taken, that they might not instantly die, for "The wages of sin is death; but the gift of God is eternal life through Jesus Christ our Lord" (Romans 6:23).
We have the benefit of hindsight, knowing what Christ's death achieved and why he had to die sacrificially, like a lamb, but before Jesus came into the world, God's people could only look forward, not knowing that Christ was the reality, "sacrificed from the founding of the world" Revelation 13:8. Earthly animal sacrifices were but shadows pointing to the eternal, heavenly reality. Only Christ has the substance. Animal sacrifices were but illustrative of him, with no need of any more after his perfect, once-for-all-time sacrifice of himself.
To give but one example of how various animals were used to deal with sin, read Leviticus chapter 4, where a young bullock, or a kid of the goats, a male, or in other cases a female, or a female lamb could all be sacrificed to deal with human sins.
This is where the Hebrew words used to say what those sacrifices symbolised needs mention in some detail. This will help us understand what Christ has actually achieved. In Revelation he is described several times as standing in the midst of God's throne in heaven, as a lamb that had been slain (past tense) yet fully alive (Revelation 5:6).
I suggested reading all of Leviticus chapter 4, where most translations say those animal sacrifices "atoned for" peoples' sins (vss. 20, 26, 31, 35). Exodus chapter 30 also speaks of an annual day of sacrifices for national sin, usually called "the Day of Atonement". Yet nowhere in the English Bible does the verb 'atone' ever occur. The Hebrew word (kippurim) is taken as expressing at-one-ment, and has been attached to the Hebrew word 'kaphar' which is also said to mean 'cover'. So, sometimes we read atonement in English and other times it is cover. But the word kaphar is not the same word as kippurim. Also, the word kaphar is repeatedly translated 'make an atonement' or similar, implying that the Hebrew consists of a phrase containing the verb 'to make', then an article, then a noun. This is not so. The original contains just the verb kaphar - supposedly meaning cover. It seems to have been decided that an 'at-one-ment' shall be 'made' but we are not told what, exactly, that means.
Also, that when this is expressed as a noun (or perhaps a participle) in a plural way, it remains singular in the English. Kaphar is a verb. It has no plural. Twice kaphar has been translated wrongly as village; it should be 'city' for it is walled - it is a containment. Kaphar, and its noun, kopher, is a matter of containment - NOT 'cover'. But the English way of rendering, 'cover' relates to the idea expressed by the English word 'forgiven'.
In Romans 5:11, Wycliffe's 1388 translation does not have the word 'atonement' (as most of our modern versions have it). He puts "by whom we have received now reconciling." This accords with vs. 10 which has 'were reconcilled'.
Yom Kippurim is the Hebrew behind the English translation of 'Day of Atonement' but that actually means the Day of Branchings'. Yom Kippurah would be the correct singular, but that never occurs in scripture, so there is no actual Day (singular) of Atonement in the Bible. Two biblical concepts have been hidden under the word 'atonement'. Kaphar, from kaph (the hand) is a matter of containment 'in hand'. It does not mean gripped, but contained in the way one would carry water to one's mouth in a cupped hand. Kaphar is strongly linked to the word 'nasa' - uplift. So, the Psalms show that David's sins were 'nasa' uplifted - and, being then uplifted from him, were kaphar - contained in hand by God - until that time when all would be fully resolved in righteousness, within Christ at Golgotha.
In Christ alone are sins uplifted, and those who find that happening to them know that their sins have been lifted up and away from them. All their sin is, by grace and faith, fully resolved in righteousness within Christ. He accomplished that at Golgotha, when sacrificed like a perfect lamb. We receive by faith the benefit of his finished work there. None of the millions of literal lambs sacrificed over the centuries could do more than point to this one, awesome sacrifice of Christ, because he is the substance; they were only ever shadows.
This is not a detour away from the question because to understand the answer, we need to know the reality of Christ as the Lamb of God that takes away the sin of the world and how literal lamb sacrifices cannot be the measure to which Christ ought to compare; it's the otherway around. No literal lamb can ever redeem from sin because atonement is not what such sacrifices were ever about. They were about symbolism for sin being lifted up and taken away. Jesus actually came down to do that, then he was lifted up, and taken back into heaven as "the Lamb of God that takes away the sin of the world" (John 1:29).
The Cherubim of Glory, Nigel Johnstone pp42-43, 155-168 snippets (Belmont Publications 2015)