Wavefunction without any basis

A wavefunction is nothing but a vector in an infinite-dimensional Hilbert space. Therefore, your question is equivalent to asking "can we represent a vector without using any basis?". The answer may be "yes" in the sense that you can write $vec{v}$ to abstractly represent the vector you are talking about. Equivalently, you can write $|psirangle$ to abstractly represent the wavefunction associated to the quantum state you are talking about. However, if you want to address the features of your vector, you need to choose a basis in order to quantitatively define them; therefore, for instance, you may choose the momentum basis to write $psi(p)=langle p|psirangle$ and $|psirangle=int_{-infty}^infty dp, psi(p)|prangle$. $psi(p)$ is what quantitatively defines the quantum object you are talking about.

Philosophically, this may be similar to the action of considering the position of a given particle in classical physics. You may say that you are talking about "the" position $mathbf{x}$ of the particle, without further defining it. However, if you want to quantitatively analyze its features, you have to choose a certain reference system and say that, for instance, the position is defined by the triplet $(5text{cm},0text{cm},-2text{cm})$ in your reference system .

Finally, note that there are some features (such as the eigenergies of the wavefunction) that are totally independent of the chosen basis, and therefore can be defined without having to rely on a certain basis.