Is there any Scripture that clearly shows that persons were baptised who were not said to believe first?
Yes, one such Scripture is Acts 16. Consider v31: “Believe on the Lord Jesus and thou shalt be saved, thou and thy house”. Now Bible students have long noted the importance of the first mention of anything in the Scriptures as being the key to its meaning throughout. The first house (or household) mentioned in the Bible is that of Noah and it also comes as part of the phrase “thou and thy house”. God said to Noah “Go into the ark, thou and all thy house”. The judgement of the flood was about to cover the earth and Noah and his house were to be saved from it. What was the basis of their salvation? Here is God’s answer: “for thee have I seen righteous before me in this generation” (Gen. 7: 1). The righteousness of one man, the head of his house, secured the whole house from judgment on the earth.
Now we have a direct parallel in the case of the jailer in Acts 16. When he was told “Believe on the Lord Jesus and thou shalt be saved, thou and thy house” (v31). the word believe is an imperative (or command) in Greek. Now imperatives are either singular or plural, depending on whether one person, or more than one person, is addressed. The Greek word here is pisteuson and (pisteuson) and it is singular. Thus salvation for the house was promised on the basis of the faith of the jailer alone. Hence, as I say, we have a parallel with Noah. In both cases, household blessing is determined by the action of the head of the house and his action alone.
Let us now consider subsequent events in Acts 16. We read “And they spoke to him the word of the Lord, with all that were in his house” (v32). The whole house heard the preaching. A little further on we read that the jailer “was baptised, he and all his straightway” (v33). Again, the whole house was baptised. Further on still we read “he laid the table [for them], and rejoiced with all his house” (v34). Yet again, the whole house rejoiced. Thus the whole house heard the preaching, rejoiced and were baptised, but who believed? Verse 34 in full reads “And having brought them into his house he laid the table [for them], and rejoiced with all his house, having believed in God”. Again, I ask the question ‘Who believed?’ While the English translation may not be absolutely clear to many, the original Greek is certain. The English words having believed translate a single Greek participle in the perfect tense: pepisteukws (pepisteukos). Participles are verbal adjectives and are thus descriptive of an action. Accordingly, in Greek they are either singular or plural and also masculine, feminine or neuter. These features identify the person who performs the action. This particular participle is masculine singular and can only refer to the jailer himself. Thus only the jailer is said to have believed. Hence we have a situation where a whole house is said to be baptised but only the head of that house is said to believe. If others had believed, Scripture could, and surely would, have said so—as it does in the case of Crispus in Acts 18: 8. In Acts 16 it doesn’t. The record of this one incident, apart from any other Scriptures, completely destroys the dogmatic view of those who insist that only believers should be baptised.
The principle of identifying a household with its head is seen throughout the Bible, not just with Noah in the OT and the Philippian jailer in the NT. Another well–known example is the night of the Passover in Egypt. The safety of the house depended on the head of the house applying the blood of the lamb to the lintel and the door–posts (see Ex. 12: 1–23). As before, the action of one man saved the house from judgment. It was “a lamb for a house” (v3). In the NT, the word of the Lord Jesus to Zacchaeus was “To–day salvation is come to this house”. What was the basis of that salvation? The answer: “inasmuch as he” (not ‘they’) “also is a son of Abraham” (Luke 19: 9, my emphasis).The Bible connects baptism with time and not eternity, with earth and not heaven, and with the kingdom and not the Church. Scripture speaks of salvation well over 300 times of which the vast majority of occasions have more to do with earth and time than heaven and eternity. While I must emphasise that eternal salvation is determined by the faith of the individual alone, there is also a practical sphere of salvation from the world connected with a believer’s household. Being brought up in such conditions involves being preserved (or saved) from many adverse moral influences, and baptism is the figure of entrance into that sphere (see 1 Pet. 3: 21). This was probably the initial thought in the word to the jailer in Acts 16: 31.