A committee of the Republican-controlled Senate yesterday rejected President Reagan's proposed elimination of the Small Business Administration in a critical early test of domestic spending cuts for next year.

The president has proposed deep cuts in such programs, and the Senate Budget Committee has approved a resolution endorsing weakened versions of some of them, but few, if any, committees have faced votes on implementing one of the his major proposals.

The action reinforced doubts that Congress will go along with the heavy cutbacks in domestic spending that Reagan has said are essential to a deficit-reduction accord with Congress that would include restraint in military spending.

In another indication of pressures driving up domestic spending, legislation was introduced in the Senate to provide about $1 billion to continue supplemental unemployment benefits for another six months. Negotiations were under way to bring some version of the measure to the floor this week, but the Senate hit a logjam over other issues, forcing it to recess until Thursday and putting its whole schedule in doubt.

Meanwhile, White House officials said presidential chief of staff Donald T. Regan is suggesting a trade-off on deficit reduction under which the senators would support bigger cuts in farm spending in exchange for more defense cuts.

"The response from up here isn't just 'no' but 'hell, no,' " said a Senate Republican source.

Reagan has proposed $6.2 billion in farm spending cuts next year, while the deficit-reduction plan approved this month by the Senate Budget Committee contemplates only about $200 million in savings from a spending freeze.

In rejecting Reagan's proposal to kill the SBA, the Senate Small Business Committee approved relatively modest cuts amounting to $879 million over three years. Reagan would save $5.3 billion over three years. The committee fell far short of the $4.7 billion in savings over the same period that was targeted by the budget committee in its deficit reduction plan.

In the 16-to-3 vote to authorize the SBA for three more years and continue most SBA spending but without further increases in funding, a majority of Republicans joined all Democrats on the panel.

While Reagan's proposal would abolish some SBA programs and move others to the Commerce Department, the committee would freeze most programs next year and allow them to grow to accommodate inflation in the two following years. It would cut direct loans to all but the handicapped, minorities and disabled veterans but would keep most other programs.

On the issue of unemployment aid, Reagan has said he opposes extending supplemental benefits, which give about 350,000 people up to 14 weeks of benefits after exhausting their state payments. The program expires Sunday.

Legislation by Sens. Arlen Specter (R-Pa.) and Carl Levin (D-Mich.) would extend the aid through the end of the fiscal year Sept. 30 while Congress works on an overhaul.

Acknowledging a "serious problem," Majority Whip Alan K. Simpson (R-Wyo.) said Senate leaders want to bring up the issue quickly. But he ran into senators who want to bring up other subjects ranging from farm credit to South Africa. The Senate hopes to work out the problem by Thursday.