Can single phase AC produce a continuous electric arc?
tl; dr version: With AC the arc current does drop to zero, but the plasma hangs around, so the visual arcing effect is continuous even if the arc current is not.

When the arc fires up it makes a plasma cloud - a region of ionized gases with lower resistance than air. This plasma persists for a time even during the AC zero-cross interval, allowing the arc jump once more at a lower strike voltage than it would in normal air. So the zero-current time is lessened in the presence of plasma for a given gap - so much so that your eye can hardly see it.

This plasma is why a ‘Jacob’s Ladder’ can work with ever-widening electrodes, even on single-phase AC: the heated plasma travels upward, making a lowest-impedance path for the arc to strike.

Here's a slow-motion video showing an AC-arc Jacob's ladder which clearly shows the arc extinguishing and relighting between cycles. Notice that the arc alternates sides as it re-strikes, while following the rising plasma cloud. https://www.youtube.com/watch?v=4zxGK0hoqIc

The mercury-vapor lamp uses a plasma arc to generate light, and they're almost always powered by single-phase AC. The lamp's arc absolutely fluctuates, but like the Jacob's ladder this isn't visible due to the human eye persistence of vision, plus the glowing of the still-excited plasma, and not the least of which the lowered strike voltage of the activated plasma. So we don't see flickering on this kind of lamp either, though its variation is measureable.

Unrectified 3-phase AC arcing behavior is the same as single-phase. Each phase has single-phase AC potential to neutral, ground, or either of two phases. So arcs phase-to-phase or phase-to-ground (or neutral, if used) will also have intervals of zero-current and cross-plasma re-strike. Likewise, so would a discharge lamp powered by two legs of a 3-phase feed.