One new pair of positive-parity chiral doublet bands have been identified in the odd-$A$ nucleus $^{135}$Nd which together with the previously reported negative-parity chiral doublet bands constitute a third case of multiple chiral doublet (M$chi$D) bands in the $Aapprox130$ mass region. The properties of the M$chi$D bands are well reproduced by constrained covariant density functional theory and particle rotor model calculations. The newly observed M$chi$D bands in $^{135}$Nd represents an important milestone in supporting the existence of M$chi$D in nuclei.
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