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To summarize some of what is stated in the comments above, there is a related question (posed as Question 4.8.5b on p. 156 of Ribes-Zalesskii) of whether torsion profinite groups have bounded exponent. As noted in the comments, your question is equivalent to this question due to Zelmanov's solution of the restricted Burnside problem, which implies that finitely generated profinite torsion groups are finite (see again p. 156 of Ribes-Zalesskii for discussion). In turn, the case of whether profinite torsion groups have bounded exponent may be reduced to the case of pro-$p$ torsion groups by a theorem of Wilson (4.8.5d in Ribes-Zalesskii, which suffices for the finitely generated case).

To summarize some of what is stated in the comments above, there is a related question (posed as Question 4.8.5b on p. 156 of Ribes-Zalesskii) of whether torsion profinite groups have bounded exponent. As noted in the comments, your question is equivalent to this question due to Zelmanov's solution of the restricted Burnside problem, which implies that finitely generated profinite torsion groups are finite (see again p. 156 of Ribes-Zalesskii for discussion). In turn, the case of whether profinite torsion groups have bounded exponent may be reduced to the case of pro-$p$ torsion groups by a theorem of Wilson (4.8.5d in Ribes-Zalesskii, which suffices for the finitely generated case).

To summarize some of what is stated in the comments above, there is a related question (posed as Question 4.8.5b on p. 156 of Ribes-Zalesskii) of whether torsion profinite groups have bounded exponent. As noted in the comments, your question is equivalent to this question due to Zelmanov's solution of the restricted Burnside problem, which implies that finitely generated profinite groups of bounded exponent are finite (see again p. 156 of Ribes-Zalesskii for discussion). In turn, the case of whether profinite torsion groups have bounded exponent may be reduced to the case of pro-$p$ torsion groups by a theorem of Wilson (4.8.5d in Ribes-Zalesskii, which suffices for the finitely generated case).

To summarize some of what is stated in the comments above, there is a related question (posed as Question 4.8.5b on p. 156 of Ribes-Zalesskii) of whether torsion profinite groups have bounded exponent. As noted in the comments, your question is equivalent to this question due to Zelmanov's solution of the restricted Burnside problem, which implies that finitely generated profinite torsion groups are finite (see again p. 156 of Ribes-Zalesskii for discussion). In turn, the case of whether profinite torsion groups have bounded exponent may be reduced to the case of pro-$p$ torsion groups by a theorem of Wilson (4.8.5d in Ribes-Zalesskii, which suffices for the finitely generated case).

To summarize some of what is stated in the comments above, there is a related question (posed as Question 4.8.5b on p. 156 of Ribes-Zalesskii) of whether torsion profinite groups have bounded exponent. As noted by @Yves Cornulier, your question is equivalent to this question due to Zelmanov's solution of the restricted Burnside problem, which implies that finitely generated profinite groups of bounded exponent are finite (see again p. 156 of Ribes-Zalesskii for discussion). In turn, the case of whether profinite torsion groups have bounded exponent may be reduced to the case of pro-$p$ torsion groups by a theorem of Wilson (4.8.5d in Ribes-Zalesskii, which suffices for the finitely generated case).

To summarize some of what is stated in the comments above, there is a related question (posed as Question 4.8.5b on p. 156 of Ribes-Zalesskii) of whether torsion profinite groups have bounded exponent. As noted in the comments, your question is equivalent to this question due to Zelmanov's solution of the restricted Burnside problem, which implies that finitely generated profinite groups of bounded exponent are finite (see again p. 156 of Ribes-Zalesskii for discussion). In turn, the case of whether profinite torsion groups have bounded exponent may be reduced to the case of pro-$p$ torsion groups by a theorem of Wilson (4.8.5d in Ribes-Zalesskii, which suffices for the finitely generated case).