We present the first Chandra and Swift X-ray study of the spiral galaxy NGC 4088 and its ultraluminous X-ray source (ULX N4088-X1). We also report very long baseline interferometry (VLBI) observations at 1.6 and 5 GHz performed quasi-simultaneously with the Swift and Chandra observations, respectively. Fifteen X-ray sources are detected by Chandra within the D25 ellipse of NGC 4088, from which we derive the X-ray luminosity function (XLF) of this galaxy. We find the XLF is very similar to those of star-forming galaxies and estimate a star-formation rate of 4.5 M ☉ yr–1. The Chandra detection of the ULX yields its most accurate X-ray position, which is spatially coincident with compact radio emission at 1.6 GHz. The ULX Chandra X-ray luminosity, L 0.2-10.0 keV = 3.4 × 1039 erg s–1, indicates that N4088-X1 could be located at the high-luminosity end of the high-mass X-ray binary (HMXB) population of NGC 4088. The estimates of the black hole (BH) mass and ratio of radio to X-ray luminosity of N4088-X1 rule out a supermassive BH nature. The Swift X-ray spectrum of N4088-X1 is best described by a thermal Comptonization model and presents a statistically significant high-energy cutoff. We conclude that N4088-X1 is most likely a stellar remnant BH in an HMXB, probably fed by Roche lobe overflow, residing in a super-Eddington ultraluminous state. The 1.6 GHz VLBI source is consistent with radio emission from possible ballistic jet ejections in this state.