Today I could prove that the existence of such family is not provable from ZFC.

**Theorem.**
If GCH holds in $V$, and $\mu\ge \omega_3$ then in $V^{Fn(\mu,2)}$ the following holds:
*If $\{A_i:i<{\omega}_3\}\subset [{\omega}_1]^{{\omega}_1}$, then there is
$I\in [\omega_3]^\omega$ and $\alpha\in \omega_3\setminus I $ such that
$$A_{\alpha}\subset \bigcup_{i\in I}A_i.$$*

**Proof.**
An ${Fn(I,2)}$-name $\dot B$ of a subset of $\omega_1$ is called
*nice* if for each ${\nu}<\omega_1$ there is an antichain
$B_{\nu}\subset {Fn(I,2)}$ such that
$$
\dot B=\{\langle p,\check{{\nu}}\rangle:{\nu}\in {\omega_1}\land p\in B_{\nu}\}=
\bigcup\{B_{\nu}\times\{\check{{\nu}}\}:{\nu}\in {\omega_1}\}.
$$
We let $supp(\dot B)=\bigcup\{dom(p):p\in\bigcup\limits_{{\nu}<{\omega_1}}B_{\nu}\}$.

It is well-known that every set of ordinals in $V[G]$ has a nice name in $V$.

If ${\varphi}$ is a bijection between two sets $I$ and $J$ then
${\varphi}$ lifts to a natural isomorphism between ${Fn(I,2)}$ and
${Fn(I,2)}$, which will be also denoted by ${\varphi}$, as follows:
for $p\in {Fn(I,2)}$ let $dom({\varphi}(p))={\varphi}''dom(p)$ and
${\varphi}(p)({\varphi}({\xi}))=p({\xi})$. Moreover ${\varphi}$ also
generates a bijection between the nice ${Fn(I,2)}$-names and the nice
$Fn(J,2)$-names:
if $\dot A$ is a nice $Fn(I,2)$-name then let
${\varphi}(\dot A )=\{\langle{\varphi}(p),\hat{{\xi}}\>:\langle p,\hat{{\xi}}\rangle \in\dot B\}$.

If $I$ and $J$ are sets of ordinals with the same order type then
${\varphi}_{I,J}$ is the natural order-preserving
bijection from $I$ onto $J$.

For each $i<{\omega}_3$ pick a nice $Fn(\mu,2)$-name $\dot A_i$
for $A_i$ in $V$. Then $S_i=supp(A_i)\in [\mu]^{\le \omega_1}$.

Since $2^{\omega_1}=\omega_2$ in $V$ there is a subset
$K\in [\omega_3]^{\omega_3}$ such that

(1) $\{S_i:i\in K\}$ forms a $\Delta$-system with kernel $S$,

(2) every element of the family $\{S_i:i\in K\}$ has the same
order type

(3) $\varphi_{S_i,S_j}$ is the identity on $S$ for all
$i,j\in K$.

(4) $\varphi_{S_i,S_J}(\dot A_i)=\dot A_j$ for
$i,j\in K$.

Let $I\in [K]^\omega$ and $\alpha \in K\setminus I$.
Assume on the contrary that $p\in Fn(\mu,2)$ and $\zeta<\omega_1$ such that
$$
p\Vdash \check \zeta\in \dot A_{\alpha}\setminus \bigcup_{i\in I}\dot A_i.
$$
Pick $\beta \in I$ such that $dom(p)\cap (S_\beta\setminus S)=\emptyset.$
Let
$$
r=p\cup \varphi_{S_\alpha, S_\beta}(p\restriction S_\alpha).
$$
Since $dom(p)\cap (S_\beta\setminus S)=\emptyset$, $r$ is a condition.
Moreover, $p\Vdash \check \zeta\in \dot A_{\alpha}$,
so $p\restriction S_\alpha \Vdash \check \zeta\in \dot A_{\alpha}$,
and so $\varphi_{S_\alpha, S_\beta}(p\restriction S_\alpha )\Vdash
\check \zeta\in \dot A_{\beta}$.

Thus $r\Vdash \zeta\in \bigcup_{i\in I}\dot A_i.$ Contradiction.