A swept source with both high repetition-rate and broad bandwidth is indispensable to enable optical coherence tomography (OCT) with high imaging rate and high axial resolution. However, available swept sources are commonly either limited in speed (sub-MHz) by inertial or kinetic component, or limited in bandwidth (<100 nm) by the gain medium. Here we report an ultrafast broadband (over 100 nm centered at 1.55-µm) all-fiber inertial-free swept source built upon a high-power dispersion-managed fiber laser in conjunction with an optical time-stretch module which bypasses complex optical amplification scheme, which result in a portable and compact implementation of time-stretch OCT (TS-OCT) at A-scan rate of 44.5-MHz, axial resolution of 14 µm in air (or 10 µm in tissue), and flat sensitivity roll-off within 4.3 mm imaging range. Together with the demonstration of two- and three-dimensional OCT imaging of a mud-fish eye anterior segment, we also perform comprehensive studies on the imaging depth, receiver bandwidth, and group velocity dispersion condition. This all-fiber inertia-free swept source could provide a promising source solution for SS-OCT system to realize high-performance volumetric OCT imaging in real time.