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This article is devoted to showing the existence and uniqueness (EU) of a solution with non-Lipschitz coefficients (NLC) of fractional Itô-Doob stochastic differential equations driven by countably many Brownian motions (FIDSDECBMs) of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ϰ</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula> by using the Picard iteration technique (PIT) and the semimartingale local time (SLT).