The$360^{\circ}$imaging has recently gained much attention; however, its angular resolution is relatively lower than that of a narrow field-of-view (FOV) perspective image as it is captured using a fisheye lens with the same sensor size. Therefore, it is beneficial to super-resolve a$360^{\circ}$image. Several attempts have been made, but mostly considered equirectangular projection (ERP) as one of the ways for$360^{\circ}$image representation despite the latitude-dependent distortions. In that case, as the output high-resolution (HR) image is always in the same ERP format as the low-resolution (LR) input, additional information loss may occur when transforming the HR image to other projection types. In this paper, we propose SphereSR, a novel framework to generate a continuous spherical image representation from an LR$360^{\circ}$image, with the goal of predicting the RGB values at given spherical coordinates for super-resolution with an arbitrary$360^{\circ}$image projection. Specifically, first we propose a feature extraction module that represents the spherical data based on an icosahedron and that efficiently extracts features on the spherical surface. We then propose a spherical local implicit image function (SLIIF) to predict RGB values at the spherical coordinates. As such, SphereSR flexibly reconstructs an HR image given an arbitrary projection type. Experiments on various benchmark datasets show that the proposed method significantly surpasses existing methods in terms of performance.