Summary: We construct a compact Hausdorff space \(K\) such that the space \(P(K)\) of Radon probabiblity measures on \(K\) considered with the weak\(^*\) topology (induced from the space of continuous functions \(C(K)\)) is countably tight which is a generalization of sequentiality (i.e., if a measure \(\mu\) is in the closure of a set \(M\), there is a countable \(M'\subseteq M\) such that \(\mu\) is in the closure of \(M'\)) but \(K\) carries a Radon probability measure which has uncountable Maharam type (i.e., \(L_1(\mu)\) is nonseparable). The construction uses (necessarily) an additional set-theoretic assumption (the \(\diamondsuit\) principle) as it was already known, by a result of Fremlin, that it is consistent that such spaces do not exist. This should be compared with the result of Plebanek and Sobota who showed that countable tightness of \(P(K\times K)\) implies that all Radon measures on \(K\) have countable type. So, our example shows that the tightness of \(P(K\times K)\) and of \(P(K)\times P(K)\) can be different as well as \(P(K)\) may have Corson property (C) while \(P(K\times K)\) fails to have it answering a question of Pol. Our construction is also a relevant example in the general context of injective tensor products of Banach spaces complementing recent results of Avilés, Martínez-Cervantes, Rodríguez and Rueda Zoca.