Let me give two examples. **Example 1.** Let us work in Gödel-Bernays set theory, and assume that $T\subset {}^{<\text{Ord}}2$ is a proper class tree of height Ord, but there is no cofinal branch. (This theory is consistent relative to an inaccessible cardinal, because if $\kappa$ is inaccessible and not weakly compact, then there is a $\kappa$-Aronszajn tree $T\subset {}^{<\kappa}2$, and then $V_\kappa$ with all subsets is a model of GBC where $T$ has the desired property.) In the logic $L_{\infty,\omega}$, which allows arbitrary sized conjunctions and disjunctions, with a constant for every element of $T$ and a unary predicate symbol $B$, consider the theory $P$ consisting of the first-order atomic diagram of $T$, together with the assertions that the elements satisfying the predicate $B$ are linearly ordered by the tree order, and furthermore, the statements for each level of the tree that precisely one element on that level is in $B$. Thus, $P$ is the theory asserting that $B$ is a cofinal branch through the tree. Every set-sized subtheory of $P$ mentions only a bounded number of levels, and so we can find a model by picking any node above that bound and using the predecessors of that node as the instantiation of $B$. But under our assumptions that the tree $T$ is Ord-Aronszajn, there can be no model of all of $P$, because such a model would determine a cofinal branch through $T$, and there is no such branch. Meanwhile, there is a strong connection between your property and (non)weak compactness, because an inaccessible cardinal $\kappa$ is weakly compact just in case we have the $\kappa$-compactness property for $L_{\kappa,\kappa}$ theories of size $\kappa$. (And there are diverse variations on this.) **Example 2.** But next, let me give a much better example, using a purely first-order language. You had alluded to the possibility that there might be no first-order example, but this isn't quite right, because of set/class issues. **Theorem.** There is a proper class first-order theory $P$, such that every set-sized subtheory of $P$ has a model, but no class is a model of the whole of $P$. **Proof.** We interpret this as a theorem scheme in ZFC, where by "class" we mean a definable class (allowing parameters). Thus, I shall provide a definition of a theory $P$, and then prove first, that every set-sized subtheory of $P$ is satisfiable, and second, that no definable class is a model of $P$. Let $P$ be the theory in the language of set theory $\in$ augmented with a predicate $\newcommand\Tr{\text{Tr}}\Tr$, meant to serve as a truth-predicate, plus a constant for every object in the universe. The theory $P$ asserts that $\Tr$ obeys all instances of the recursive Tarskian truth definition: - $\Tr(a\in b)$ just in case $a\in b$ holds. - $\Tr(\varphi\wedge\psi)$ just in case $\Tr(\varphi)$ and $\Tr(\psi)$. - $\Tr(\neg\varphi)$ just in case $\Tr(\varphi)$ does not hold. - $\Tr(\exists x\ \varphi)$ just in case there is $a$ such that $\Tr(\varphi(a))$. For any set many such assertions, we can find a model, since we can find some $V_\theta$ large enough to contain all the parameters mentioned in the subtheory, and then use truth in $\langle V_\theta,\in\rangle$, which will satisfy all the assertions made in the subtheory. But no definable class can satisfy $P$, because this is exactly the content of Tarski's theorem on the non-definability of truth. **QED**