OTTAWA—More asylum seekers are crossing illegally into Canada from the U.S., many trekking across snow-covered fields along unguarded stretches of the border, in the wake of the Trump administration’s push to tighten immigration rules.

At least 450 people crossed illegally in January and February, despite bitter cold that usually discourages such traffic, government officials said Thursday. In all of last year, the total was 2,400.

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