This is question has a long and interesting history, which is discussed in Arnie Miller's paper cited below. The first construction of a model of ZFC + $\aleph_1 < 2^{\aleph_0}$ where $[0,1]$ can be partitioned into $\aleph_1$ pairwise disjoint nonempty closed sets is due to Jim Baumgartner (unpublished).

Early on, Hausdorff showed that one can always write $[0,1]$ as a union of $\aleph_1$ pairwise disjoint nonempty $F_{\sigma\delta}$ sets.

*Hausdorff, F.*, **Summen von $\aleph_1$ Mengen.**, Fundam. Math., Warszawa, 26, 241-255 (1936). ZBL62.0228.03.

Sierpiński then asked if $F_{\sigma\delta}$ could be improved by $G_\delta$ in Hausdorff's result.

*Sierpiński, Wacław*, **Sur deux consequences d’un théorème de Hausdorff**, Fundam. Math. 33, 269-272 (1945). ZBL0060.12715.

Fremlin and Shelah answered Sierpiński's question by showing that $[0,1]$ can be written as a union of $\aleph_1$ pairwise disjoint nonempty $G_\delta$ (equivalently $G_{\delta\sigma}$) sets if and only if $[0,1]$ can be covered by $\aleph_1$ meager sets.

*Fremlin, D. H.; Shelah, S.*, **On partitions of the real line**, Isr. J. Math. 32, 299-304 (1979). ZBL0413.04002.

Then, Miller showed that the Fremlin-Shelah result is false if we replace $G_\delta$ by closed (equivalently $F_\sigma$).

*Miller, Arnold W.*, Covering $2^\omega$ with $\omega_1$ disjoint closed sets, The Kleene Symp., Proc., Madison/Wis. 1978, Stud. Logic Found. Math., Vol. 101, 415-421 (1980). ZBL0444.03026. http://www.math.wisc.edu/~miller/res/cov.pdf

However, it is still consistent with $\aleph_1 < 2^{\aleph_0}$ that $[0,1]$ can be written as a union of $\aleph_1$ pairwise disjoint nonempty closed sets. Baumgartner was perhaps the first to find such a model (see Theorem 4 in Miller's paper), but this result was rediscovered many times (with different arguments). For example, the earliest published proof is due to Stern, who showed that this is true in any forcing extension of a model of CH by adding $\aleph_2$ random reals.

*Stern, Jacques*, Partitions of the real line into $\aleph_1$ closed sets, Higher Set Theory, Proc. Oberwolfach 1977, Lect. Notes Math. 669, 455-460 (1978). ZBL0393.03038.

It is perhaps interesting that even without assuming AC, one can always partition $[0,1]$ into $\aleph_1$ pairwise disjoint nonempty Borel sets. (See this MO answer by Andreas Blass.) Hausdorff's gap construction makes heavy use of choice, so one can ask whether it is provable in ZF that one can always partition $[0,1]$ into $\aleph_1$ pairwise disjoint nonempty Borel sets of bounded rank. This question was investigated by Stern. The answer is negative, assuming the consistency of some large cardinal axioms. For example, the answer is no assuming AD.

*Stern, Jacques*, **Effective partitions of the real line into Borel sets of bounded rank**, Ann. Math. Logic 18, 29-60 (1980). ZBL0522.03032.