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Caldera unrest is often caused by kilometer‐sized kilometer‐deep sills. Still unanswered questions include the following: How do sills spread? Why can magma propagate for kilometers without solidifying? Why do ground deformation data rarely, if ever, detect sill propagation? We show that kilometer‐sized kilometer‐deep magmatic sills spread like hydraulic fractures in an infinite medium. How magma propagates depends on overburden pressure, magma viscosity, injection rate, and difference between magma and rock temperatures. A small lag, filled with vapors from the fluid and/or the rock, exists between the propagating magma and fracture fronts. If the sill spreads along an interface, the lag slightly affects isothermal sill spreading but takes a key role in the case of nonisothermal propagation: A sill would stop after few tens of meters without it, unless magma intrudes rocks that are as hot as the solidification temperature or has unrealistic overpressures, because spreading velocity decreases soon to the critical value at which the tip becomes blocked with solidified magma. The lag defers magma solidification as heat exchange between the magma and the rock is effective only behind the thermal‐insulating lag, where magma has some finite thickness and sill opening grows with distance from the tip faster than thickness of solidified magma. Thus, the critical velocity decreases, allowing greater maximum sill sizes. We also show that the ground deformation pattern does not change appreciably over time if the final sill radius is smaller than 2 to 3 km, explaining why deformation is usually attributed to the inflation of a stationary source. Plain Language Summary: Quick eruption of large magma volumes may cause subsidence and form a caldera. Although calderas are particularly active dangerous volcanoes, many urban areas are located nearby or within them. Calderas often suffer unrest, defined by ground deformation, enhanced seismicity, and increased degassing: Sometimes unrest leads to eruption, at other times it does not. Caldera unrest is often attributed to the emplacement and inflation of a kilometer‐sized flat magmatic intrusion (sill) at few‐kilometer depth; remarkable examples are Fernandina and Sierra Negra (Galápagos), Kilauea south caldera (Hawaii), and Campi Flegrei (Italy). The mechanism controlling magmatic sill formation is under investigation since at least half a century, but important questions are still unanswered, among which are the following: How do sills spread? Why can magma propagate for kilometers without solidifying? Why do ground deformation data rarely, if ever, detect sill propagation, while they usually evidence inflation? We modeled sill spreading and found that kilometer‐sized sills do not require high rock temperatures and/or injection rates to form: A small thermal‐insulating lag, located between the propagating magma and fracture fronts and filled with vapors from the magma and/or the rock, delays magma solidification and allows sill spreading for kilometers. Key Points: Kilometer‐sized kilometer‐deep magmatic sills spread in homogeneous rock or along a rock interface like hydraulic fractures in an infinite mediumThe small lag between the magma and fracture fronts slightly affects spreading velocity but greatly defers stopping by magma solidificationFor sill radii shorter than 2–3 km, the ground deformation pattern is about constant during spreading, as for a stationary inflating source [ABSTRACT FROM AUTHOR]