DUBAI, United Arab Emirates — At Lifestyle, an inexpensive home furnishings store tucked inside the Mall of the Emirates here, shoppers on Wednesday examined lamps and picked up picture frames.

But they did not see any Trump Home mirrors or lotion dispensers.

Just hours after the Republican presidential candidate Donald J. Trump called for a ban on the entry of Muslims into the United States, one of his Middle Eastern business partners, the Dubai-based Landmark Group, responded by removing all Trump-branded products from the shelves of its Lifestyle retail stores.

In the Middle East, as in other parts of the world, the foundation of Mr. Trump’s business is his name, a brand that for many there is synonymous with American success and luxury. Regarding the Middle East as a critical avenue for growth, the Trump Organization signed numerous deals in recent years to put the Trump brand on upscale golf courses and residences in Dubai, hotel properties in Turkey, and home furnishings, accessories and décor sold in stores across the region.

But the value of the Trump name in the world’s vast Muslim population has been thrown into question since he made his comments.