This work presents the construction of a novel spherical wavelet basis designed for incomplete spherical datasets, i.e. datasets which are missing in a particular region of the sphere. The eigenfunctions of the Slepian spatial-spectral concentration problem (the Slepian functions) are a set of orthogonal basis functions which are more concentrated within a defined region. Slepian functions allow one to compute a convolution on the incomplete sphere by leveraging the recently proposed sifting convolution and extending it to any set of basis functions. Through a tiling of the Slepian harmonic line, one may construct scale-discretised wavelets. An illustration is presented based on an example region on the sphere defined by the topographic map of the Earth. The Slepian wavelets and corresponding wavelet coefficients are constructed from this region and are used in a straightforward denoising example.